Systems of Polynomial Equations - SAT Math
Card 1 of 87
Which method of solving is suitable for non-linear systems?
Which method of solving is suitable for non-linear systems?
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Substitution or elimination. Both methods work by reducing the system to simpler equations.
Substitution or elimination. Both methods work by reducing the system to simpler equations.
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What is the solution to the system $x + y = 5$ and $x - y = 1$?
What is the solution to the system $x + y = 5$ and $x - y = 1$?
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$(x, y) = (3, 2)$. Add equations to get $2x = 4$, then substitute back.
$(x, y) = (3, 2)$. Add equations to get $2x = 4$, then substitute back.
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What is the solution to $x^2 + 6x + 9 = 0$?
What is the solution to $x^2 + 6x + 9 = 0$?
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$x = -3$. Perfect square trinomial $(x + 3)^2 = 0$.
$x = -3$. Perfect square trinomial $(x + 3)^2 = 0$.
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Solve for $y$: $y^2 - 4y + 4 = 0$.
Solve for $y$: $y^2 - 4y + 4 = 0$.
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$y = 2$. Perfect square trinomial with double root.
$y = 2$. Perfect square trinomial with double root.
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What is the general form of a system of polynomial equations?
What is the general form of a system of polynomial equations?
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$f(x, y) = 0, , g(x, y) = 0$. Two polynomial equations set equal to zero that must be solved simultaneously.
$f(x, y) = 0, , g(x, y) = 0$. Two polynomial equations set equal to zero that must be solved simultaneously.
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What is the degree of $4x^5 - x^3 + 2x - 1$?
What is the degree of $4x^5 - x^3 + 2x - 1$?
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$5$. Highest power determines polynomial degree.
$5$. Highest power determines polynomial degree.
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What is the solution to $y = x^2 - 1$ and $y = 0$?
What is the solution to $y = x^2 - 1$ and $y = 0$?
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$x = 1$ or $x = -1$. Set $x^2 - 1 = 0$ and factor.
$x = 1$ or $x = -1$. Set $x^2 - 1 = 0$ and factor.
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What is the vertex of $y = (x - 1)^2 - 4$?
What is the vertex of $y = (x - 1)^2 - 4$?
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$(1, -4)$. Vertex form shows vertex at $(h, k)$.
$(1, -4)$. Vertex form shows vertex at $(h, k)$.
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Identify the solution of the system: $x^2 + y = 5$ and $x + y^2 = 5$.
Identify the solution of the system: $x^2 + y = 5$ and $x + y^2 = 5$.
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$(1, 2)$ and $(2, 1)$. Both points satisfy both equations when substituted.
$(1, 2)$ and $(2, 1)$. Both points satisfy both equations when substituted.
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What is the general form of a quadratic equation?
What is the general form of a quadratic equation?
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$ax^2 + bx + c = 0$. Standard form with $a \neq 0$.
$ax^2 + bx + c = 0$. Standard form with $a \neq 0$.
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Is the system $x^2 + y^2 = 1$ and $x + y = 1$ linear or non-linear?
Is the system $x^2 + y^2 = 1$ and $x + y = 1$ linear or non-linear?
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Non-linear. The presence of $x^2 + y^2$ makes the system non-linear.
Non-linear. The presence of $x^2 + y^2$ makes the system non-linear.
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What is the degree of the polynomial $2x^3 - 4x^2 + x - 5$?
What is the degree of the polynomial $2x^3 - 4x^2 + x - 5$?
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- Highest power of the variable in the polynomial.
- Highest power of the variable in the polynomial.
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Identify the leading coefficient of $3x^4 - x^3 + 2x^2$.
Identify the leading coefficient of $3x^4 - x^3 + 2x^2$.
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- Coefficient of the highest degree term.
- Coefficient of the highest degree term.
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What is the degree of the polynomial $5x^4 - 3x^2 + 2$?
What is the degree of the polynomial $5x^4 - 3x^2 + 2$?
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- Highest power of the variable term.
- Highest power of the variable term.
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What is the solution to $x^2 + 1 = 0$?
What is the solution to $x^2 + 1 = 0$?
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$x = i$ or $x = -i$. Complex roots when discriminant is negative.
$x = i$ or $x = -i$. Complex roots when discriminant is negative.
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What is the leading term of $5x^3 + 4x^2 - x$?
What is the leading term of $5x^3 + 4x^2 - x$?
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$5x^3$. Term with highest degree in the polynomial.
$5x^3$. Term with highest degree in the polynomial.
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What is the degree of a system consisting of $x^3 + y^3 = 7$ and $xy = 1$?
What is the degree of a system consisting of $x^3 + y^3 = 7$ and $xy = 1$?
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Degree 3. The highest degree among all terms in the system is 3.
Degree 3. The highest degree among all terms in the system is 3.
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Solve: $x^2 - 2x - 8 = 0$.
Solve: $x^2 - 2x - 8 = 0$.
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$x = 4$ or $x = -2$. Factor as $(x - 4)(x + 2) = 0$.
$x = 4$ or $x = -2$. Factor as $(x - 4)(x + 2) = 0$.
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What is the $y$-intercept of $y = 3x + 4$?
What is the $y$-intercept of $y = 3x + 4$?
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- Value where line crosses the $y$-axis.
- Value where line crosses the $y$-axis.
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Solve: $x^2 - 5x + 6 = 0$.
Solve: $x^2 - 5x + 6 = 0$.
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$x = 3$ or $x = 2$. Factor as $(x - 3)(x - 2) = 0$.
$x = 3$ or $x = 2$. Factor as $(x - 3)(x - 2) = 0$.
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Find a common solution to $x^2 - 4x + 4 = 0$ and $y = x - 2$.
Find a common solution to $x^2 - 4x + 4 = 0$ and $y = x - 2$.
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$(2, 0)$. First equation gives $x = 2$, then substitute into second equation.
$(2, 0)$. First equation gives $x = 2$, then substitute into second equation.
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Find the $x$-intercepts of $y = x^2 - 16$.
Find the $x$-intercepts of $y = x^2 - 16$.
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$x = 4$ or $x = -4$. Set $y = 0$ and solve $x^2 - 16 = 0$.
$x = 4$ or $x = -4$. Set $y = 0$ and solve $x^2 - 16 = 0$.
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What is the factored form of $x^2 - 9$?
What is the factored form of $x^2 - 9$?
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$(x - 3)(x + 3)$. Difference of squares pattern: $a^2 - b^2$.
$(x - 3)(x + 3)$. Difference of squares pattern: $a^2 - b^2$.
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Find the $y$-intercept of $y = x^2 + 3x + 2$.
Find the $y$-intercept of $y = x^2 + 3x + 2$.
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- Substitute $x = 0$ into the equation.
- Substitute $x = 0$ into the equation.
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Convert $x^2 - 4x + 4$ to vertex form.
Convert $x^2 - 4x + 4$ to vertex form.
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$(x - 2)^2$. Perfect square trinomial in factored form.
$(x - 2)^2$. Perfect square trinomial in factored form.
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What is the quadratic formula?
What is the quadratic formula?
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$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Standard formula for solving quadratic equations.
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Standard formula for solving quadratic equations.
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What is the sum of the roots of $ax^2 + bx + c = 0$?
What is the sum of the roots of $ax^2 + bx + c = 0$?
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$-\frac{b}{a}$. From Vieta's formulas for quadratic equations.
$-\frac{b}{a}$. From Vieta's formulas for quadratic equations.
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Solve: $y = 2x + 3$ and $y = -x + 1$.
Solve: $y = 2x + 3$ and $y = -x + 1$.
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$(x, y) = (-\frac{2}{3}, \frac{5}{3})$. Set equations equal: $2x + 3 = -x + 1$.
$(x, y) = (-\frac{2}{3}, \frac{5}{3})$. Set equations equal: $2x + 3 = -x + 1$.
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Solve: $x^2 + y^2 = 25$ and $x = 3y$.
Solve: $x^2 + y^2 = 25$ and $x = 3y$.
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$(x, y) = (3, 1)$ or $(x, y) = (-3, -1)$. Substitute $x = 3y$ into circle equation.
$(x, y) = (3, 1)$ or $(x, y) = (-3, -1)$. Substitute $x = 3y$ into circle equation.
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Identify the roots of $x^2 - 4x = 0$.
Identify the roots of $x^2 - 4x = 0$.
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$x = 0$ or $x = 4$. Factor out $x$: $x(x - 4) = 0$.
$x = 0$ or $x = 4$. Factor out $x$: $x(x - 4) = 0$.
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Solve for $x$: $x^2 + y^2 = 13$ and $x + y = 3$.
Solve for $x$: $x^2 + y^2 = 13$ and $x + y = 3$.
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$x = 2$ or $x = 1$. Substitute $y = 3 - x$ into first equation and solve.
$x = 2$ or $x = 1$. Substitute $y = 3 - x$ into first equation and solve.
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What is the solution to $y = 3x - 2$ and $y = -x + 6$?
What is the solution to $y = 3x - 2$ and $y = -x + 6$?
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$(x, y) = (2, 4)$. Set equations equal: $3x - 2 = -x + 6$.
$(x, y) = (2, 4)$. Set equations equal: $3x - 2 = -x + 6$.
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What is the definition of a polynomial?
What is the definition of a polynomial?
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An expression of finite terms with non-negative integer exponents. Sum of terms with variables raised to whole number powers.
An expression of finite terms with non-negative integer exponents. Sum of terms with variables raised to whole number powers.
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What is the factored form of $x^2 + 5x + 6$?
What is the factored form of $x^2 + 5x + 6$?
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$(x + 2)(x + 3)$. Find two numbers that multiply to 6 and add to 5.
$(x + 2)(x + 3)$. Find two numbers that multiply to 6 and add to 5.
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Solve the system: $x^2 + y^2 = 25$ and $y = 3x$. Find one solution.
Solve the system: $x^2 + y^2 = 25$ and $y = 3x$. Find one solution.
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$(3, 9)$. Substitute $y = 3x$ into the first equation to find $x = 3$.
$(3, 9)$. Substitute $y = 3x$ into the first equation to find $x = 3$.
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What method can be used to solve a system of polynomial equations graphically?
What method can be used to solve a system of polynomial equations graphically?
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Graph the equations and find intersection points. Solutions occur where the curves visually cross on the coordinate plane.
Graph the equations and find intersection points. Solutions occur where the curves visually cross on the coordinate plane.
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What is the formula for the discriminant of a quadratic equation $ax^2 + bx + c = 0$?
What is the formula for the discriminant of a quadratic equation $ax^2 + bx + c = 0$?
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$b^2 - 4ac$. Determines nature of roots for quadratic equations.
$b^2 - 4ac$. Determines nature of roots for quadratic equations.
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What is the vertex of $y = x^2 + 6x + 9$?
What is the vertex of $y = x^2 + 6x + 9$?
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$(-3, 0)$. Perfect square $(x + 3)^2$ has vertex at $x = -3$.
$(-3, 0)$. Perfect square $(x + 3)^2$ has vertex at $x = -3$.
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Identify the constant term in $3x^3 - 4x + 5$.
Identify the constant term in $3x^3 - 4x + 5$.
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- Term without any variable.
- Term without any variable.
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What are the solutions to $x^2 = 9$?
What are the solutions to $x^2 = 9$?
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$x = 3$ or $x = -3$. Take square root of both sides.
$x = 3$ or $x = -3$. Take square root of both sides.
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Solve for $x$: $x^2 + 4x + 4 = 0$.
Solve for $x$: $x^2 + 4x + 4 = 0$.
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$x = -2$. Perfect square trinomial $(x + 2)^2 = 0$.
$x = -2$. Perfect square trinomial $(x + 2)^2 = 0$.
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What do the solutions of a system of polynomial equations represent?
What do the solutions of a system of polynomial equations represent?
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Intersection points of the graphs. Each solution represents a point where all equations are satisfied.
Intersection points of the graphs. Each solution represents a point where all equations are satisfied.
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What is the product of the roots of $ax^2 + bx + c = 0$?
What is the product of the roots of $ax^2 + bx + c = 0$?
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$\frac{c}{a}$. From Vieta's formulas for quadratic equations.
$\frac{c}{a}$. From Vieta's formulas for quadratic equations.
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Find the vertex form of $y = x^2 - 4x + 4$.
Find the vertex form of $y = x^2 - 4x + 4$.
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$y = (x - 2)^2$. Complete the square to get vertex form.
$y = (x - 2)^2$. Complete the square to get vertex form.
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What is the factored form of $x^2 - 2x - 3$?
What is the factored form of $x^2 - 2x - 3$?
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$ (x - 3)(x + 1) $. Factor by finding two numbers that multiply to $-3$.
$ (x - 3)(x + 1) $. Factor by finding two numbers that multiply to $-3$.
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What is the solution to $x^2 + 2x + 1 = 0$?
What is the solution to $x^2 + 2x + 1 = 0$?
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$x = -1$. Perfect square trinomial $(x + 1)^2 = 0$.
$x = -1$. Perfect square trinomial $(x + 1)^2 = 0$.
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Solve: $y = x^2 - 4$ and $y = 0$.
Solve: $y = x^2 - 4$ and $y = 0$.
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$x = 2$ or $x = -2$. Set $x^2 - 4 = 0$ and solve.
$x = 2$ or $x = -2$. Set $x^2 - 4 = 0$ and solve.
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Solve the system: $2x + 3y = 7$ and $4x - y = 1$.
Solve the system: $2x + 3y = 7$ and $4x - y = 1$.
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$(x, y) = (1, \frac{5}{3})$. Solve by elimination or substitution method.
$(x, y) = (1, \frac{5}{3})$. Solve by elimination or substitution method.
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What is the product of the roots of $ax^2 + bx + c = 0$?
What is the product of the roots of $ax^2 + bx + c = 0$?
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$\frac{c}{a}$. From Vieta's formulas for quadratic equations.
$\frac{c}{a}$. From Vieta's formulas for quadratic equations.
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What is the sum of the roots of $ax^2 + bx + c = 0$?
What is the sum of the roots of $ax^2 + bx + c = 0$?
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$-\frac{b}{a}$. From Vieta's formulas for quadratic equations.
$-\frac{b}{a}$. From Vieta's formulas for quadratic equations.
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What is the solution to $y = 3x - 2$ and $y = -x + 6$?
What is the solution to $y = 3x - 2$ and $y = -x + 6$?
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$(x, y) = (2, 4)$. Set equations equal: $3x - 2 = -x + 6$.
$(x, y) = (2, 4)$. Set equations equal: $3x - 2 = -x + 6$.
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What is the solution to the system $x + y = 5$ and $x - y = 1$?
What is the solution to the system $x + y = 5$ and $x - y = 1$?
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$(x, y) = (3, 2)$. Add equations to get $2x = 4$, then substitute back.
$(x, y) = (3, 2)$. Add equations to get $2x = 4$, then substitute back.
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What is the solution to $x^2 + 1 = 0$?
What is the solution to $x^2 + 1 = 0$?
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$x = i$ or $x = -i$. Complex roots when discriminant is negative.
$x = i$ or $x = -i$. Complex roots when discriminant is negative.
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Solve for $y$: $y^2 - 4y + 4 = 0$.
Solve for $y$: $y^2 - 4y + 4 = 0$.
Tap to reveal answer
$y = 2$. Perfect square trinomial with double root.
$y = 2$. Perfect square trinomial with double root.
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What is the factored form of $x^2 - 9$?
What is the factored form of $x^2 - 9$?
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$(x - 3)(x + 3)$. Difference of squares pattern: $a^2 - b^2$.
$(x - 3)(x + 3)$. Difference of squares pattern: $a^2 - b^2$.
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Identify the leading coefficient of $3x^4 - x^3 + 2x^2$.
Identify the leading coefficient of $3x^4 - x^3 + 2x^2$.
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- Coefficient of the highest degree term.
- Coefficient of the highest degree term.
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What is the vertex of $y = (x - 1)^2 - 4$?
What is the vertex of $y = (x - 1)^2 - 4$?
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$(1, -4)$. Vertex form shows vertex at $(h, k)$.
$(1, -4)$. Vertex form shows vertex at $(h, k)$.
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Solve: $x^2 - 2x - 8 = 0$.
Solve: $x^2 - 2x - 8 = 0$.
Tap to reveal answer
$x = 4$ or $x = -2$. Factor as $(x - 4)(x + 2) = 0$.
$x = 4$ or $x = -2$. Factor as $(x - 4)(x + 2) = 0$.
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What is the quadratic formula?
What is the quadratic formula?
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$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Standard formula for solving quadratic equations.
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Standard formula for solving quadratic equations.
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Identify the constant term in $3x^3 - 4x + 5$.
Identify the constant term in $3x^3 - 4x + 5$.
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- Term without any variable.
- Term without any variable.
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Solve: $x^2 - 5x + 6 = 0$.
Solve: $x^2 - 5x + 6 = 0$.
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$x = 3$ or $x = 2$. Factor as $(x - 3)(x - 2) = 0$.
$x = 3$ or $x = 2$. Factor as $(x - 3)(x - 2) = 0$.
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What is the degree of $4x^5 - x^3 + 2x - 1$?
What is the degree of $4x^5 - x^3 + 2x - 1$?
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- Highest power determines polynomial degree.
- Highest power determines polynomial degree.
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What is the factored form of $x^2 - 2x - 3$?
What is the factored form of $x^2 - 2x - 3$?
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$(x - 3)(x + 1)$. Factor by finding two numbers that multiply to $-3$.
$(x - 3)(x + 1)$. Factor by finding two numbers that multiply to $-3$.
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Solve: $y = x^2 - 4$ and $y = 0$.
Solve: $y = x^2 - 4$ and $y = 0$.
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$x = 2$ or $x = -2$. Set $x^2 - 4 = 0$ and solve.
$x = 2$ or $x = -2$. Set $x^2 - 4 = 0$ and solve.
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Find the $x$-intercepts of $y = x^2 - 16$.
Find the $x$-intercepts of $y = x^2 - 16$.
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$x = 4$ or $x = -4$. Set $y = 0$ and solve $x^2 - 16 = 0$.
$x = 4$ or $x = -4$. Set $y = 0$ and solve $x^2 - 16 = 0$.
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What is the solution to $y = x^2 - 1$ and $y = 0$?
What is the solution to $y = x^2 - 1$ and $y = 0$?
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$x = 1$ or $x = -1$. Set $x^2 - 1 = 0$ and factor.
$x = 1$ or $x = -1$. Set $x^2 - 1 = 0$ and factor.
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Solve for $x$: $x^2 + 4x + 4 = 0$.
Solve for $x$: $x^2 + 4x + 4 = 0$.
Tap to reveal answer
$x = -2$. Perfect square trinomial $(x + 2)^2 = 0$.
$x = -2$. Perfect square trinomial $(x + 2)^2 = 0$.
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What is the $y$-intercept of $y = 3x + 4$?
What is the $y$-intercept of $y = 3x + 4$?
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$4$. Value where line crosses the $y$-axis.
$4$. Value where line crosses the $y$-axis.
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What is the factored form of $x^2 + 5x + 6$?
What is the factored form of $x^2 + 5x + 6$?
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$(x + 2)(x + 3)$. Find two numbers that multiply to 6 and add to 5.
$(x + 2)(x + 3)$. Find two numbers that multiply to 6 and add to 5.
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What is the solution to $x^2 + 2x + 1 = 0$?
What is the solution to $x^2 + 2x + 1 = 0$?
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$x = -1$. Perfect square trinomial $(x + 1)^2 = 0$.
$x = -1$. Perfect square trinomial $(x + 1)^2 = 0$.
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Find the $y$-intercept of $y = x^2 + 3x + 2$.
Find the $y$-intercept of $y = x^2 + 3x + 2$.
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- Substitute $x = 0$ into the equation.
- Substitute $x = 0$ into the equation.
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Solve: $x^2 + y^2 = 25$ and $x = 3y$.
Solve: $x^2 + y^2 = 25$ and $x = 3y$.
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$(x, y) = (3, 1)$ or $(x, y) = (-3, -1)$. Substitute $x = 3y$ into circle equation.
$(x, y) = (3, 1)$ or $(x, y) = (-3, -1)$. Substitute $x = 3y$ into circle equation.
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Find the vertex form of $y = x^2 - 4x + 4$.
Find the vertex form of $y = x^2 - 4x + 4$.
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$y = (x - 2)^2$. Complete the square to get vertex form.
$y = (x - 2)^2$. Complete the square to get vertex form.
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What is the degree of the polynomial $5x^4 - 3x^2 + 2$?
What is the degree of the polynomial $5x^4 - 3x^2 + 2$?
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- Highest power of the variable term.
- Highest power of the variable term.
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Convert $x^2 - 4x + 4$ to vertex form.
Convert $x^2 - 4x + 4$ to vertex form.
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$(x - 2)^2$. Perfect square trinomial in factored form.
$(x - 2)^2$. Perfect square trinomial in factored form.
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What is the formula for the discriminant of a quadratic equation $ax^2 + bx + c = 0$?
What is the formula for the discriminant of a quadratic equation $ax^2 + bx + c = 0$?
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$b^2 - 4ac$. Determines nature of roots for quadratic equations.
$b^2 - 4ac$. Determines nature of roots for quadratic equations.
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Solve the system: $2x + 3y = 7$ and $4x - y = 1$.
Solve the system: $2x + 3y = 7$ and $4x - y = 1$.
Tap to reveal answer
$(x, y) = (1, \frac{5}{3})$. Solve by elimination or substitution method.
$(x, y) = (1, \frac{5}{3})$. Solve by elimination or substitution method.
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What is the solution to $x^2 + 6x + 9 = 0$?
What is the solution to $x^2 + 6x + 9 = 0$?
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$x = -3$. Perfect square trinomial $(x + 3)^2 = 0$.
$x = -3$. Perfect square trinomial $(x + 3)^2 = 0$.
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What is the vertex of $y = x^2 + 6x + 9$?
What is the vertex of $y = x^2 + 6x + 9$?
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$(-3, 0)$. Perfect square $(x + 3)^2$ has vertex at $x = -3$.
$(-3, 0)$. Perfect square $(x + 3)^2$ has vertex at $x = -3$.
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What is the leading term of $5x^3 + 4x^2 - x$?
What is the leading term of $5x^3 + 4x^2 - x$?
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$5x^3$. Term with highest degree in the polynomial.
$5x^3$. Term with highest degree in the polynomial.
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Identify the roots of $x^2 - 4x = 0$.
Identify the roots of $x^2 - 4x = 0$.
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$x = 0$ or $x = 4$. Factor out $x$: $x(x - 4) = 0$.
$x = 0$ or $x = 4$. Factor out $x$: $x(x - 4) = 0$.
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What is the degree of the polynomial $2x^3 - 4x^2 + x - 5$?
What is the degree of the polynomial $2x^3 - 4x^2 + x - 5$?
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- Highest power of the variable in the polynomial.
- Highest power of the variable in the polynomial.
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Solve for $x$: $x^2 + y^2 = 13$ and $x + y = 3$.
Solve for $x$: $x^2 + y^2 = 13$ and $x + y = 3$.
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$x = 2$ or $x = 1$. Substitute $y = 3 - x$ into first equation and solve.
$x = 2$ or $x = 1$. Substitute $y = 3 - x$ into first equation and solve.
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Solve: $y = 2x + 3$ and $y = -x + 1$.
Solve: $y = 2x + 3$ and $y = -x + 1$.
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$(x, y) = (-\frac{2}{3}, \frac{5}{3})$. Set equations equal: $2x + 3 = -x + 1$.
$(x, y) = (-\frac{2}{3}, \frac{5}{3})$. Set equations equal: $2x + 3 = -x + 1$.
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What is the general form of a quadratic equation?
What is the general form of a quadratic equation?
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$ax^2 + bx + c = 0$. Standard form with $a \neq 0$.
$ax^2 + bx + c = 0$. Standard form with $a \neq 0$.
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What is the definition of a polynomial?
What is the definition of a polynomial?
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An expression of finite terms with non-negative integer exponents. Sum of terms with variables raised to whole number powers.
An expression of finite terms with non-negative integer exponents. Sum of terms with variables raised to whole number powers.
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What are the solutions to $x^2 = 9$?
What are the solutions to $x^2 = 9$?
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$x = 3$ or $x = -3$. Take square root of both sides.
$x = 3$ or $x = -3$. Take square root of both sides.
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