Solving Nonlinear Functions - PSAT Math
Card 1 of 30
What is the solution of $x^2=50$ in simplest radical form?
What is the solution of $x^2=50$ in simplest radical form?
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$x=\pm 5\sqrt{2}$. Take square root of both sides: $x=\pm\sqrt{50}=\pm\sqrt{25\cdot 2}=\pm 5\sqrt{2}$.
$x=\pm 5\sqrt{2}$. Take square root of both sides: $x=\pm\sqrt{50}=\pm\sqrt{25\cdot 2}=\pm 5\sqrt{2}$.
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What is the zero-product property used to solve $(x-p)(x-q)=0$?
What is the zero-product property used to solve $(x-p)(x-q)=0$?
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$x=p$ or $x=q$. If a product equals zero, at least one factor must be zero.
$x=p$ or $x=q$. If a product equals zero, at least one factor must be zero.
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What is the solution of $x^2-9=0$?
What is the solution of $x^2-9=0$?
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$x=\pm 3$. Factor as $(x-3)(x+3)=0$, then apply zero-product property.
$x=\pm 3$. Factor as $(x-3)(x+3)=0$, then apply zero-product property.
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What are the solutions of $(x-4)(x+1)=0$?
What are the solutions of $(x-4)(x+1)=0$?
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$x=4$ and $x=-1$. Apply zero-product property: each factor equals zero.
$x=4$ and $x=-1$. Apply zero-product property: each factor equals zero.
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What is the solution set of $x^2+5x=0$?
What is the solution set of $x^2+5x=0$?
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$x=0$ and $x=-5$. Factor out $x$: $x(x+5)=0$, so $x=0$ or $x=-5$.
$x=0$ and $x=-5$. Factor out $x$: $x(x+5)=0$, so $x=0$ or $x=-5$.
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What are the solutions of $x^2-5x+6=0$?
What are the solutions of $x^2-5x+6=0$?
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$x=2$ and $x=3$. Factors to $(x-2)(x-3)=0$; apply zero-product property.
$x=2$ and $x=3$. Factors to $(x-2)(x-3)=0$; apply zero-product property.
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What is the solution set of $|x-3|=5$?
What is the solution set of $|x-3|=5$?
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$x=8$ and $x=-2$. Split into two cases: $x-3=5$ or $x-3=-5$.
$x=8$ and $x=-2$. Split into two cases: $x-3=5$ or $x-3=-5$.
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What is the solution of $\frac{1}{x}=3$?
What is the solution of $\frac{1}{x}=3$?
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$x=\frac{1}{3}$. Multiply both sides by $x$: $1=3x$, so $x=\frac{1}{3}$.
$x=\frac{1}{3}$. Multiply both sides by $x$: $1=3x$, so $x=\frac{1}{3}$.
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What is the solution of $2x^2-8x=0$?
What is the solution of $2x^2-8x=0$?
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$x=0$ and $x=4$. Factor out $2x$: $2x(x-4)=0$, giving $x=0$ or $x=4$.
$x=0$ and $x=4$. Factor out $2x$: $2x(x-4)=0$, giving $x=0$ or $x=4$.
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What is the solution set of $\sqrt{x+2}=x$?
What is the solution set of $\sqrt{x+2}=x$?
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$x=2$. Square both sides: $x+2=x^2$; solve $x^2-x-2=0$ to get $x=2$ (reject $x=-1$).
$x=2$. Square both sides: $x+2=x^2$; solve $x^2-x-2=0$ to get $x=2$ (reject $x=-1$).
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What is the solution of $\sqrt{x-1}=4$?
What is the solution of $\sqrt{x-1}=4$?
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$x=17$. Square both sides: $x-1=16$, so $x=17$.
$x=17$. Square both sides: $x-1=16$, so $x=17$.
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Solve for $x$: $\frac{x+1}{x-2}=3$ and state any restriction on $x$.
Solve for $x$: $\frac{x+1}{x-2}=3$ and state any restriction on $x$.
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$x=\frac{7}{2}$, with $x\ne 2$. Cross-multiply: $x+1=3(x-2)$, then solve.
$x=\frac{7}{2}$, with $x\ne 2$. Cross-multiply: $x+1=3(x-2)$, then solve.
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What is the zero-product property used after factoring?
What is the zero-product property used after factoring?
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If $ab=0$, then $a=0$ or $b=0$. At least one factor must be zero for the product to be zero.
If $ab=0$, then $a=0$ or $b=0$. At least one factor must be zero for the product to be zero.
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Solve for $x$: $x^2+6x+9=0$.
Solve for $x$: $x^2+6x+9=0$.
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$x=-3$. Perfect square: $(x+3)^2=0$ gives a double root.
$x=-3$. Perfect square: $(x+3)^2=0$ gives a double root.
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Solve for $x$: $x^{\frac{1}{2}}=3$.
Solve for $x$: $x^{\frac{1}{2}}=3$.
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$x=9$. Square both sides: $(x^{\frac{1}{2}})^2=3^2$.
$x=9$. Square both sides: $(x^{\frac{1}{2}})^2=3^2$.
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What are the solutions to $x^2+6x+9=0$?
What are the solutions to $x^2+6x+9=0$?
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$x=-3$. Perfect square $(x+3)^2=0$ has one repeated root.
$x=-3$. Perfect square $(x+3)^2=0$ has one repeated root.
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What is the solution set of $\frac{2}{x-1}=1$?
What is the solution set of $\frac{2}{x-1}=1$?
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$x=3$. Cross-multiply: $2=x-1$, so $x=3$.
$x=3$. Cross-multiply: $2=x-1$, so $x=3$.
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What substitution is used to solve a quadratic in $x^2$, such as $x^4+5x^2+4=0$?
What substitution is used to solve a quadratic in $x^2$, such as $x^4+5x^2+4=0$?
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Let $u=x^2$. Transforms to $u^2+5u+4=0$, a standard quadratic.
Let $u=x^2$. Transforms to $u^2+5u+4=0$, a standard quadratic.
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Identify the required restriction when solving $\frac{p(x)}{q(x)}=r$.
Identify the required restriction when solving $\frac{p(x)}{q(x)}=r$.
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Exclude values where $q(x)=0$. Division by zero is undefined, so check denominator.
Exclude values where $q(x)=0$. Division by zero is undefined, so check denominator.
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Solve for $x$: $x^4-5x^2+4=0$.
Solve for $x$: $x^4-5x^2+4=0$.
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$x=\pm 1$ or $x=\pm 2$. Let $u=x^2$: $u^2-5u+4=0$ factors to $(u-1)(u-4)=0$.
$x=\pm 1$ or $x=\pm 2$. Let $u=x^2$: $u^2-5u+4=0$ factors to $(u-1)(u-4)=0$.
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What is the correct first step to solve a rational equation like $\frac{p(x)}{q(x)}=r$?
What is the correct first step to solve a rational equation like $\frac{p(x)}{q(x)}=r$?
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Multiply both sides by $q(x)$, with $q(x)\ne 0$. Clears the denominator to create a polynomial equation.
Multiply both sides by $q(x)$, with $q(x)\ne 0$. Clears the denominator to create a polynomial equation.
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What is the standard method to solve $a(bx+c)^2=d$ when $a\ne 0$ and $d/a\ge 0$?
What is the standard method to solve $a(bx+c)^2=d$ when $a\ne 0$ and $d/a\ge 0$?
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$(bx+c)=\pm\sqrt{\frac{d}{a}}$. Isolate the squared term, then take square root of both sides.
$(bx+c)=\pm\sqrt{\frac{d}{a}}$. Isolate the squared term, then take square root of both sides.
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Identify the key step to solve $x^2=k$ for real $x$ when $k\ge 0$.
Identify the key step to solve $x^2=k$ for real $x$ when $k\ge 0$.
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$x=\pm\sqrt{k}$. Take square root of both sides, considering both positive and negative roots.
$x=\pm\sqrt{k}$. Take square root of both sides, considering both positive and negative roots.
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What is the quadratic formula for solving $ax^2+bx+c=0$ in terms of $a$, $b$, and $c$?
What is the quadratic formula for solving $ax^2+bx+c=0$ in terms of $a$, $b$, and $c$?
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$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived using completing the square on the general quadratic equation.
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived using completing the square on the general quadratic equation.
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What is the discriminant of $ax^2+bx+c=0$, used to classify the number of real solutions?
What is the discriminant of $ax^2+bx+c=0$, used to classify the number of real solutions?
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$\Delta=b^2-4ac$. Appears under the square root in the quadratic formula.
$\Delta=b^2-4ac$. Appears under the square root in the quadratic formula.
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What does $\Delta>0$ imply about the real solutions of $ax^2+bx+c=0$?
What does $\Delta>0$ imply about the real solutions of $ax^2+bx+c=0$?
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$2$ distinct real solutions. Positive discriminant means the square root yields two different values.
$2$ distinct real solutions. Positive discriminant means the square root yields two different values.
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What does $\Delta=0$ imply about the real solutions of $ax^2+bx+c=0$?
What does $\Delta=0$ imply about the real solutions of $ax^2+bx+c=0$?
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$1$ real solution (a repeated root). Zero discriminant makes $\pm\sqrt{0}=0$, giving one solution.
$1$ real solution (a repeated root). Zero discriminant makes $\pm\sqrt{0}=0$, giving one solution.
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What does $\Delta<0$ imply about the real solutions of $ax^2+bx+c=0$?
What does $\Delta<0$ imply about the real solutions of $ax^2+bx+c=0$?
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No real solutions. Negative discriminant means no real square root exists.
No real solutions. Negative discriminant means no real square root exists.
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What is the axis of symmetry of $y=ax^2+bx+c$ written in terms of $a$ and $b$?
What is the axis of symmetry of $y=ax^2+bx+c$ written in terms of $a$ and $b$?
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$x=-\frac{b}{2a}$. The parabola's line of symmetry passes through the vertex.
$x=-\frac{b}{2a}$. The parabola's line of symmetry passes through the vertex.
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What is the vertex $x$-coordinate of $y=ax^2+bx+c$ in terms of $a$ and $b$?
What is the vertex $x$-coordinate of $y=ax^2+bx+c$ in terms of $a$ and $b$?
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$x=-\frac{b}{2a}$. The vertex lies on the axis of symmetry.
$x=-\frac{b}{2a}$. The vertex lies on the axis of symmetry.
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