Systems of Polynomial Equations - SAT Math
Card 1 of 39
What is the product of the roots of $ax^2 + bx + c = 0$?
What is the product of the roots of $ax^2 + bx + c = 0$?
Tap to reveal answer
$\frac{c}{a}$. From Vieta's formulas for quadratic equations.
$\frac{c}{a}$. From Vieta's formulas for quadratic equations.
← Didn't Know|Knew It →
What is the sum of the roots of $ax^2 + bx + c = 0$?
What is the sum of the roots of $ax^2 + bx + c = 0$?
Tap to reveal answer
$-\frac{b}{a}$. From Vieta's formulas for quadratic equations.
$-\frac{b}{a}$. From Vieta's formulas for quadratic equations.
← Didn't Know|Knew It →
What is the solution to $x^2 + 1 = 0$?
What is the solution to $x^2 + 1 = 0$?
Tap to reveal answer
$x = i$ or $x = -i$. Complex roots when discriminant is negative.
$x = i$ or $x = -i$. Complex roots when discriminant is negative.
← Didn't Know|Knew It →
What is the solution to $y = 3x - 2$ and $y = -x + 6$?
What is the solution to $y = 3x - 2$ and $y = -x + 6$?
Tap to reveal answer
$(x, y) = (2, 4)$. Set equations equal: $3x - 2 = -x + 6$.
$(x, y) = (2, 4)$. Set equations equal: $3x - 2 = -x + 6$.
← Didn't Know|Knew It →
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 $x - y = 1$?
Tap to reveal answer
$(x, y) = (3, 2)$. Add equations to get $2x = 4$, then substitute back.
$(x, y) = (3, 2)$. Add equations to get $2x = 4$, then substitute back.
← Didn't Know|Knew It →
What are the solutions to $x^2 = 9$?
What are the solutions to $x^2 = 9$?
Tap to reveal answer
$x = 3$ or $x = -3$. Take square root of both sides.
$x = 3$ or $x = -3$. Take square root of both sides.
← Didn't Know|Knew It →
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.
← Didn't Know|Knew It →
What is the factored form of $x^2 - 9$?
What is the factored form of $x^2 - 9$?
Tap to reveal answer
$(x - 3)(x + 3)$. Difference of squares pattern: $a^2 - b^2$.
$(x - 3)(x + 3)$. Difference of squares pattern: $a^2 - b^2$.
← Didn't Know|Knew It →
Identify the leading coefficient of $3x^4 - x^3 + 2x^2$.
Identify the leading coefficient of $3x^4 - x^3 + 2x^2$.
Tap to reveal answer
- Coefficient of the highest degree term.
- Coefficient of the highest degree term.
← Didn't Know|Knew It →
What is the vertex of $y = (x - 1)^2 - 4$?
What is the vertex of $y = (x - 1)^2 - 4$?
Tap to reveal answer
$(1, -4)$. Vertex form shows vertex at $(h, k)$.
$(1, -4)$. Vertex form shows vertex at $(h, k)$.
← Didn't Know|Knew It →
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 $(x - 4)(x + 2) = 0$.
← Didn't Know|Knew It →
What is the quadratic formula?
What is the quadratic formula?
Tap to reveal answer
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Standard formula for solving quadratic equations.
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Standard formula for solving quadratic equations.
← Didn't Know|Knew It →
Identify the constant term in $3x^3 - 4x + 5$.
Identify the constant term in $3x^3 - 4x + 5$.
Tap to reveal answer
- Term without any variable.
- Term without any variable.
← Didn't Know|Knew It →
Solve: $x^2 - 5x + 6 = 0$.
Solve: $x^2 - 5x + 6 = 0$.
Tap to reveal answer
$x = 3$ or $x = 2$. Factor as $(x - 3)(x - 2) = 0$.
$x = 3$ or $x = 2$. Factor as $(x - 3)(x - 2) = 0$.
← Didn't Know|Knew It →
What is the degree of $4x^5 - x^3 + 2x - 1$?
What is the degree of $4x^5 - x^3 + 2x - 1$?
Tap to reveal answer
- Highest power determines polynomial degree.
- Highest power determines polynomial degree.
← Didn't Know|Knew It →
What is the factored form of $x^2 - 2x - 3$?
What is the factored form of $x^2 - 2x - 3$?
Tap to reveal answer
$(x - 3)(x + 1)$. Factor by finding two numbers that multiply to $-3$.
$(x - 3)(x + 1)$. Factor by finding two numbers that multiply to $-3$.
← Didn't Know|Knew It →
Solve: $y = x^2 - 4$ and $y = 0$.
Solve: $y = x^2 - 4$ and $y = 0$.
Tap to reveal answer
$x = 2$ or $x = -2$. Set $x^2 - 4 = 0$ and solve.
$x = 2$ or $x = -2$. Set $x^2 - 4 = 0$ and solve.
← Didn't Know|Knew It →
Find the $x$-intercepts of $y = x^2 - 16$.
Find the $x$-intercepts of $y = x^2 - 16$.
Tap to reveal answer
$x = 4$ or $x = -4$. Set $y = 0$ and solve $x^2 - 16 = 0$.
$x = 4$ or $x = -4$. Set $y = 0$ and solve $x^2 - 16 = 0$.
← Didn't Know|Knew It →
What is the solution to $y = x^2 - 1$ and $y = 0$?
What is the solution to $y = x^2 - 1$ and $y = 0$?
Tap to reveal answer
$x = 1$ or $x = -1$. Set $x^2 - 1 = 0$ and factor.
$x = 1$ or $x = -1$. Set $x^2 - 1 = 0$ and factor.
← Didn't Know|Knew It →
Solve for $x$: $x^2 + y^2 = 13$ and $x + y = 3$.
Solve for $x$: $x^2 + y^2 = 13$ and $x + y = 3$.
Tap to reveal answer
$x = 2$ 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.
← Didn't Know|Knew It →
What is the $y$-intercept of $y = 3x + 4$?
What is the $y$-intercept of $y = 3x + 4$?
Tap to reveal answer
- Value where line crosses the $y$-axis.
- Value where line crosses the $y$-axis.
← Didn't Know|Knew It →
What is the factored form of $x^2 + 5x + 6$?
What is the factored form of $x^2 + 5x + 6$?
Tap to reveal answer
$(x + 2)(x + 3)$. 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.
← Didn't Know|Knew It →
What is the solution to $x^2 + 2x + 1 = 0$?
What is the solution to $x^2 + 2x + 1 = 0$?
Tap to reveal answer
$x = -1$. Perfect square trinomial $(x + 1)^2 = 0$.
$x = -1$. Perfect square trinomial $(x + 1)^2 = 0$.
← Didn't Know|Knew It →
Find the $y$-intercept of $y = x^2 + 3x + 2$.
Find the $y$-intercept of $y = x^2 + 3x + 2$.
Tap to reveal answer
- Substitute $x = 0$ into the equation.
- Substitute $x = 0$ into the equation.
← Didn't Know|Knew It →
Solve: $x^2 + y^2 = 25$ and $x = 3y$.
Solve: $x^2 + y^2 = 25$ and $x = 3y$.
Tap to reveal answer
$(x, y) = (3, 1)$ 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.
← Didn't Know|Knew It →
Find the vertex form of $y = x^2 - 4x + 4$.
Find the vertex form of $y = x^2 - 4x + 4$.
Tap to reveal answer
$y = (x - 2)^2$. Complete the square to get vertex form.
$y = (x - 2)^2$. Complete the square to get vertex form.
← Didn't Know|Knew It →
What is the degree of the polynomial $5x^4 - 3x^2 + 2$?
What is the degree of the polynomial $5x^4 - 3x^2 + 2$?
Tap to reveal answer
- Highest power of the variable term.
- Highest power of the variable term.
← Didn't Know|Knew It →
Convert $x^2 - 4x + 4$ to vertex form.
Convert $x^2 - 4x + 4$ to vertex form.
Tap to reveal answer
$(x - 2)^2$. Perfect square trinomial in factored form.
$(x - 2)^2$. Perfect square trinomial in factored form.
← Didn't Know|Knew It →
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 $ax^2 + bx + c = 0$?
Tap to reveal answer
$b^2 - 4ac$. Determines nature of roots for quadratic equations.
$b^2 - 4ac$. Determines nature of roots for quadratic equations.
← Didn't Know|Knew It →
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.
← Didn't Know|Knew It →
What is the solution to $x^2 + 6x + 9 = 0$?
What is the solution to $x^2 + 6x + 9 = 0$?
Tap to reveal answer
$x = -3$. Perfect square trinomial $(x + 3)^2 = 0$.
$x = -3$. Perfect square trinomial $(x + 3)^2 = 0$.
← Didn't Know|Knew It →
What is the vertex of $y = x^2 + 6x + 9$?
What is the vertex of $y = x^2 + 6x + 9$?
Tap to reveal answer
$(-3, 0)$. Perfect square $(x + 3)^2$ has vertex at $x = -3$.
$(-3, 0)$. Perfect square $(x + 3)^2$ has vertex at $x = -3$.
← Didn't Know|Knew It →
What is the leading term of $5x^3 + 4x^2 - x$?
What is the leading term of $5x^3 + 4x^2 - x$?
Tap to reveal answer
$5x^3$. Term with highest degree in the polynomial.
$5x^3$. Term with highest degree in the polynomial.
← Didn't Know|Knew It →
Identify the roots of $x^2 - 4x = 0$.
Identify the roots of $x^2 - 4x = 0$.
Tap to reveal answer
$x = 0$ or $x = 4$. Factor out $x$: $x(x - 4) = 0$.
$x = 0$ or $x = 4$. Factor out $x$: $x(x - 4) = 0$.
← Didn't Know|Knew It →
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 $(x + 2)^2 = 0$.
← Didn't Know|Knew It →
What is the degree of the polynomial $2x^3 - 4x^2 + x - 5$?
What is the degree of the polynomial $2x^3 - 4x^2 + x - 5$?
Tap to reveal answer
- Highest power of the variable in the polynomial.
- Highest power of the variable in the polynomial.
← Didn't Know|Knew It →
Solve: $y = 2x + 3$ and $y = -x + 1$.
Solve: $y = 2x + 3$ and $y = -x + 1$.
Tap to reveal answer
$(x, y) = (-\frac{2}{3}, \frac{5}{3})$. Set equations equal: $2x + 3 = -x + 1$.
$(x, y) = (-\frac{2}{3}, \frac{5}{3})$. Set equations equal: $2x + 3 = -x + 1$.
← Didn't Know|Knew It →
What is the general form of a quadratic equation?
What is the general form of a quadratic equation?
Tap to reveal answer
$ax^2 + bx + c = 0$. Standard form with $a \neq 0$.
$ax^2 + bx + c = 0$. Standard form with $a \neq 0$.
← Didn't Know|Knew It →
What is the definition of a polynomial?
What is the definition of a polynomial?
Tap to reveal answer
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.
← Didn't Know|Knew It →