Complete the Square to Find Solutions - Algebra 2
Card 1 of 30
Solve by completing the square: $x^2+2x-7=0$.
Solve by completing the square: $x^2+2x-7=0$.
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$x=-1\pm 2\sqrt{2}$. From $(x+1)^2=8$, take square root: $x=-1\pm 2\sqrt{2}$.
$x=-1\pm 2\sqrt{2}$. From $(x+1)^2=8$, take square root: $x=-1\pm 2\sqrt{2}$.
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What are the solutions of $(x-p)^2=q$ written explicitly?
What are the solutions of $(x-p)^2=q$ written explicitly?
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$x=p\pm\sqrt{q}$. Explicit form after isolating $x$.
$x=p\pm\sqrt{q}$. Explicit form after isolating $x$.
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Rewrite $x^2+9x$ as a square plus a constant: $x^2+9x=\left(x+__\right)^2-__$.
Rewrite $x^2+9x$ as a square plus a constant: $x^2+9x=\left(x+__\right)^2-__$.
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$\left(x+\frac{9}{2}\right)^2-\frac{81}{4}$. Half of $9$ is $\frac{9}{2}$, then subtract $\left(\frac{9}{2}\right)^2$.
$\left(x+\frac{9}{2}\right)^2-\frac{81}{4}$. Half of $9$ is $\frac{9}{2}$, then subtract $\left(\frac{9}{2}\right)^2$.
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Transform $2x^2+8x+3=0$ into $(x-p)^2=q$ form.
Transform $2x^2+8x+3=0$ into $(x-p)^2=q$ form.
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$\left(x+2\right)^2=\frac{5}{2}$. First divide by $2$, then complete the square.
$\left(x+2\right)^2=\frac{5}{2}$. First divide by $2$, then complete the square.
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What is the quadratic formula for solutions to $ax^2+bx+c=0$?
What is the quadratic formula for solutions to $ax^2+bx+c=0$?
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$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square on the general form.
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square on the general form.
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What expression is the discriminant in $ax^2+bx+c=0$?
What expression is the discriminant in $ax^2+bx+c=0$?
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$b^2-4ac$. Determines the nature of quadratic solutions.
$b^2-4ac$. Determines the nature of quadratic solutions.
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What does $b^2-4ac>0$ tell you about the solutions of $ax^2+bx+c=0$?
What does $b^2-4ac>0$ tell you about the solutions of $ax^2+bx+c=0$?
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Two distinct real solutions. Positive discriminant means two real intersection points.
Two distinct real solutions. Positive discriminant means two real intersection points.
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What does $b^2-4ac=0$ tell you about the solutions of $ax^2+bx+c=0$?
What does $b^2-4ac=0$ tell you about the solutions of $ax^2+bx+c=0$?
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One real double root. Zero discriminant means one repeated real solution.
One real double root. Zero discriminant means one repeated real solution.
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What does $b^2-4ac<0$ tell you about the solutions of $ax^2+bx+c=0$?
What does $b^2-4ac<0$ tell you about the solutions of $ax^2+bx+c=0$?
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No real solutions (two complex solutions). Negative discriminant means no real intersection points.
No real solutions (two complex solutions). Negative discriminant means no real intersection points.
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What is the vertex form obtained by completing the square for $y=ax^2+bx+c$?
What is the vertex form obtained by completing the square for $y=ax^2+bx+c$?
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$y=a(x-h)^2+k$. Standard vertex form from completing the square.
$y=a(x-h)^2+k$. Standard vertex form from completing the square.
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What is $h$ in vertex form $y=a(x-h)^2+k$ in terms of $a$ and $b$?
What is $h$ in vertex form $y=a(x-h)^2+k$ in terms of $a$ and $b$?
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$h=-\frac{b}{2a}$. Formula for the x-coordinate of the vertex.
$h=-\frac{b}{2a}$. Formula for the x-coordinate of the vertex.
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What is the key identity used to expand $(x+p)^2$ while completing the square?
What is the key identity used to expand $(x+p)^2$ while completing the square?
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$(x+p)^2=x^2+2px+p^2$. Fundamental binomial expansion used in completing squares.
$(x+p)^2=x^2+2px+p^2$. Fundamental binomial expansion used in completing squares.
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What operation do you use after writing a quadratic as $(x-p)^2=q$ to solve for $x$?
What operation do you use after writing a quadratic as $(x-p)^2=q$ to solve for $x$?
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Take square roots: $x-p=\pm\sqrt{q}$. Square root both sides to solve for $x$.
Take square roots: $x-p=\pm\sqrt{q}$. Square root both sides to solve for $x$.
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What is the completing-the-square step after getting $x^2+bx$ on one side?
What is the completing-the-square step after getting $x^2+bx$ on one side?
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Add $\left(\frac{b}{2}\right)^2$ to both sides. Maintains equation balance while creating a perfect square.
Add $\left(\frac{b}{2}\right)^2$ to both sides. Maintains equation balance while creating a perfect square.
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What is the first step to complete the square in $ax^2+bx+c=0$ when $a\neq 1$?
What is the first step to complete the square in $ax^2+bx+c=0$ when $a\neq 1$?
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Divide by $a$ to make the $x^2$ coefficient $1$. Makes the leading coefficient 1 for easier completion.
Divide by $a$ to make the $x^2$ coefficient $1$. Makes the leading coefficient 1 for easier completion.
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What perfect square trinomial equals $x^2+bx+\left(\frac{b}{2}\right)^2$?
What perfect square trinomial equals $x^2+bx+\left(\frac{b}{2}\right)^2$?
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$\left(x+\frac{b}{2}\right)^2$. The completed perfect square trinomial form.
$\left(x+\frac{b}{2}\right)^2$. The completed perfect square trinomial form.
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What value is added to $x^2+bx$ to complete the square?
What value is added to $x^2+bx$ to complete the square?
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$\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$, then squared.
$\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$, then squared.
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Identify the value added to complete the square in $x^2-10x+__$.
Identify the value added to complete the square in $x^2-10x+__$.
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$25$. Half of $-10$ is $-5$, squared gives $25$.
$25$. Half of $-10$ is $-5$, squared gives $25$.
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What are the solutions of $(x-p)^2=q$ written explicitly?
What are the solutions of $(x-p)^2=q$ written explicitly?
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$x=p\pm\sqrt{q}$. Explicit form after isolating $x$.
$x=p\pm\sqrt{q}$. Explicit form after isolating $x$.
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What does the symbol $\pm$ indicate when solving $(x-p)^2=q$?
What does the symbol $\pm$ indicate when solving $(x-p)^2=q$?
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Two cases: $+$ and $-$ square roots. Plus-minus accounts for both positive and negative square roots.
Two cases: $+$ and $-$ square roots. Plus-minus accounts for both positive and negative square roots.
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What is the goal form when completing the square for a quadratic in $x$?
What is the goal form when completing the square for a quadratic in $x$?
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$(x-p)^2 = q$. This standard form isolates the squared term and constant.
$(x-p)^2 = q$. This standard form isolates the squared term and constant.
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Rewrite $x^2-5x$ as a square plus a constant: $x^2-5x=\left(x-__\right)^2-__$.
Rewrite $x^2-5x$ as a square plus a constant: $x^2-5x=\left(x-__\right)^2-__$.
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$\left(x-\frac{5}{2}\right)^2-\frac{25}{4}$. Half of $ -5 $ is $ -\frac{5}{2} $, then subtract $ \left(\frac{5}{2}\right)^2 $.
$\left(x-\frac{5}{2}\right)^2-\frac{25}{4}$. Half of $ -5 $ is $ -\frac{5}{2} $, then subtract $ \left(\frac{5}{2}\right)^2 $.
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What square expression forms from $x^2+\frac{b}{a}x+\left(\frac{b}{2a}\right)^2$?
What square expression forms from $x^2+\frac{b}{a}x+\left(\frac{b}{2a}\right)^2$?
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$\left(x+\frac{b}{2a}\right)^2$. The perfect square trinomial after completing.
$\left(x+\frac{b}{2a}\right)^2$. The perfect square trinomial after completing.
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After $x^2+\frac{b}{a}x=-\frac{c}{a}$, what is added to both sides to complete the square?
After $x^2+\frac{b}{a}x=-\frac{c}{a}$, what is added to both sides to complete the square?
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$\left(\frac{b}{2a}\right)^2$. Half the new coefficient of $x$, then squared.
$\left(\frac{b}{2a}\right)^2$. Half the new coefficient of $x$, then squared.
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What is the result after dividing $ax^2+bx+c=0$ by $a$?
What is the result after dividing $ax^2+bx+c=0$ by $a$?
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$x^2+\frac{b}{a}x+\frac{c}{a}=0$. Standard form after dividing by the leading coefficient.
$x^2+\frac{b}{a}x+\frac{c}{a}=0$. Standard form after dividing by the leading coefficient.
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Solve by completing the square: $4x^2+4x-3=0$.
Solve by completing the square: $4x^2+4x-3=0$.
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$x=\frac{1}{2}$ or $x=-\frac{3}{2}$. From $(x+\frac{1}{2})^2=1$, solve: $x=-\frac{1}{2}\pm 1$.
$x=\frac{1}{2}$ or $x=-\frac{3}{2}$. From $(x+\frac{1}{2})^2=1$, solve: $x=-\frac{1}{2}\pm 1$.
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Transform $4x^2+4x-3=0$ into $(x-p)^2=q$ form.
Transform $4x^2+4x-3=0$ into $(x-p)^2=q$ form.
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$\left(x+\frac{1}{2}\right)^2=1$. First divide by $4$, then complete the square.
$\left(x+\frac{1}{2}\right)^2=1$. First divide by $4$, then complete the square.
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Solve by completing the square: $3x^2-12x+1=0$.
Solve by completing the square: $3x^2-12x+1=0$.
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$x=2\pm\frac{\sqrt{33}}{3}$. From $(x-2)^2=\frac{11}{3}$, solve: $x=2\pm\sqrt{\frac{11}{3}}$.
$x=2\pm\frac{\sqrt{33}}{3}$. From $(x-2)^2=\frac{11}{3}$, solve: $x=2\pm\sqrt{\frac{11}{3}}$.
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Transform $3x^2-12x+1=0$ into $(x-p)^2=q$ form.
Transform $3x^2-12x+1=0$ into $(x-p)^2=q$ form.
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$\left(x-2\right)^2=\frac{11}{3}$. First divide by $3$, then complete the square.
$\left(x-2\right)^2=\frac{11}{3}$. First divide by $3$, then complete the square.
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Solve by completing the square: $2x^2+8x+3=0$.
Solve by completing the square: $2x^2+8x+3=0$.
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$x=-2\pm\frac{\sqrt{10}}{2}$. From $(x+2)^2=\frac{5}{2}$, solve: $x=-2\pm\sqrt{\frac{5}{2}}$.
$x=-2\pm\frac{\sqrt{10}}{2}$. From $(x+2)^2=\frac{5}{2}$, solve: $x=-2\pm\sqrt{\frac{5}{2}}$.
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