Complete the Square to Find Solutions - Algebra
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What is the completed-square form of $2x^2-4x-1=0$?
What is the completed-square form of $2x^2-4x-1=0$?
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$\left(x-1\right)^2=\frac{3}{2}$. First divide by 2: $x^2-2x-\frac{1}{2}=0$, then complete the square.
$\left(x-1\right)^2=\frac{3}{2}$. First divide by 2: $x^2-2x-\frac{1}{2}=0$, then complete the square.
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What equation results just before taking square roots when deriving the quadratic formula?
What equation results just before taking square roots when deriving the quadratic formula?
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$\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}$. This form leads directly to the quadratic formula when square roots are taken.
$\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}$. This form leads directly to the quadratic formula when square roots are taken.
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What expression results after completing the square on $x^2+\frac{b}{a}x$?
What expression results after completing the square on $x^2+\frac{b}{a}x$?
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$\left(x+\frac{b}{2a}\right)^2$. This is the perfect square formed after adding $\left(\frac{b}{2a}\right)^2$.
$\left(x+\frac{b}{2a}\right)^2$. This is the perfect square formed after adding $\left(\frac{b}{2a}\right)^2$.
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What is the key added term after dividing by $a$ in $ax^2+bx+c=0$ to complete the square?
What is the key added term after dividing by $a$ in $ax^2+bx+c=0$ to complete the square?
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$\left(\frac{b}{2a}\right)^2$. After dividing by $a$, add half of the new linear coefficient squared.
$\left(\frac{b}{2a}\right)^2$. After dividing by $a$, add half of the new linear coefficient squared.
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What are the solutions of $2x^2-4x-1=0$?
What are the solutions of $2x^2-4x-1=0$?
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$x=1\pm\frac{\sqrt{6}}{2}$. From $(x-1)^2=\frac{3}{2}$: $x=1\pm\sqrt{\frac{3}{2}}=1\pm\frac{\sqrt{6}}{2}$.
$x=1\pm\frac{\sqrt{6}}{2}$. From $(x-1)^2=\frac{3}{2}$: $x=1\pm\sqrt{\frac{3}{2}}=1\pm\frac{\sqrt{6}}{2}$.
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What are the solutions of $5x^2+20x+15=0$?
What are the solutions of $5x^2+20x+15=0$?
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$x=-1$ or $x=-3$. From $(x+2)^2=1$: $x+2=\pm 1$, so $x=-2\pm 1$.
$x=-1$ or $x=-3$. From $(x+2)^2=1$: $x+2=\pm 1$, so $x=-2\pm 1$.
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What is the completed-square form of $5x^2+20x+15=0$?
What is the completed-square form of $5x^2+20x+15=0$?
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$\left(x+2\right)^2=1$. First divide by 5: $x^2+4x+3=0$, then complete the square.
$\left(x+2\right)^2=1$. First divide by 5: $x^2+4x+3=0$, then complete the square.
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What are the solutions of $x^2-2x-8=0$?
What are the solutions of $x^2-2x-8=0$?
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$x=4$ or $x=-2$. From $(x-1)^2=9$: $x-1=\pm 3$, so $x=1\pm 3$.
$x=4$ or $x=-2$. From $(x-1)^2=9$: $x-1=\pm 3$, so $x=1\pm 3$.
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What is the completed-square form of $x^2-2x-8=0$?
What is the completed-square form of $x^2-2x-8=0$?
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$\left(x-1\right)^2=9$. Completing the square: $(x-1)^2=1+8=9$.
$\left(x-1\right)^2=9$. Completing the square: $(x-1)^2=1+8=9$.
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What are the solutions of $x^2+6x+13=0$?
What are the solutions of $x^2+6x+13=0$?
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$x=-3\pm 2i$. From $(x+3)^2=-4$: $x=-3\pm\sqrt{-4}=-3\pm 2i$.
$x=-3\pm 2i$. From $(x+3)^2=-4$: $x=-3\pm\sqrt{-4}=-3\pm 2i$.
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What is the completed-square form of $x^2+6x+13=0$?
What is the completed-square form of $x^2+6x+13=0$?
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$\left(x+3\right)^2=-4$. Completing the square: $(x+3)^2=9-13=-4$.
$\left(x+3\right)^2=-4$. Completing the square: $(x+3)^2=9-13=-4$.
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What is $p$ and $q$ if $x^2-6x+1=0$ is written as $\left(x-p\right)^2=q$?
What is $p$ and $q$ if $x^2-6x+1=0$ is written as $\left(x-p\right)^2=q$?
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$p=3,\ q=8$. From $x^2-6x+1=0$: $(x-3)^2=9-1=8$.
$p=3,\ q=8$. From $x^2-6x+1=0$: $(x-3)^2=9-1=8$.
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What is the missing term to complete the square: $x^2-\frac{7}{3}x+\square=\left(x-\frac{7}{6}\right)^2$?
What is the missing term to complete the square: $x^2-\frac{7}{3}x+\square=\left(x-\frac{7}{6}\right)^2$?
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$\frac{49}{36}$. The missing term is $\left(\frac{-7/3}{2}\right)^2=\left(\frac{-7}{6}\right)^2=\frac{49}{36}$.
$\frac{49}{36}$. The missing term is $\left(\frac{-7/3}{2}\right)^2=\left(\frac{-7}{6}\right)^2=\frac{49}{36}$.
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What is the missing term to complete the square: $x^2+\frac{5}{2}x+\square=\left(x+\frac{5}{4}\right)^2$?
What is the missing term to complete the square: $x^2+\frac{5}{2}x+\square=\left(x+\frac{5}{4}\right)^2$?
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$\frac{25}{16}$. The missing term is $\left(\frac{5/2}{2}\right)^2=\left(\frac{5}{4}\right)^2=\frac{25}{16}$.
$\frac{25}{16}$. The missing term is $\left(\frac{5/2}{2}\right)^2=\left(\frac{5}{4}\right)^2=\frac{25}{16}$.
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What is the missing term to complete the square: $x^2-12x+\square=\left(x-6\right)^2$?
What is the missing term to complete the square: $x^2-12x+\square=\left(x-6\right)^2$?
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$36$. The missing term is $\left(\frac{-12}{2}\right)^2=(-6)^2=36$.
$36$. The missing term is $\left(\frac{-12}{2}\right)^2=(-6)^2=36$.
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What conclusion about real solutions follows from $\left(x+1\right)^2=-9$?
What conclusion about real solutions follows from $\left(x+1\right)^2=-9$?
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No real solutions. A perfect square cannot equal a negative real number.
No real solutions. A perfect square cannot equal a negative real number.
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What is the completed-square form of $x^2+2x+10=0$?
What is the completed-square form of $x^2+2x+10=0$?
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$\left(x+1\right)^2=-9$. Completing the square gives a perfect square equal to a negative number.
$\left(x+1\right)^2=-9$. Completing the square gives a perfect square equal to a negative number.
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What are the solutions of $4x^2+4x-3=0$?
What are the solutions of $4x^2+4x-3=0$?
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$x=\frac{1}{2}$ or $x=-\frac{3}{2}$. From $(x+\frac{1}{2})^2=1$: $x=-\frac{1}{2}\pm 1$.
$x=\frac{1}{2}$ or $x=-\frac{3}{2}$. From $(x+\frac{1}{2})^2=1$: $x=-\frac{1}{2}\pm 1$.
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What is the completed-square form of $4x^2+4x-3=0$?
What is the completed-square form of $4x^2+4x-3=0$?
Tap to reveal answer
$\left(x+\frac{1}{2}\right)^2=1$. First divide by 4, then complete the square on $x^2+x-\frac{3}{4}=0$.
$\left(x+\frac{1}{2}\right)^2=1$. First divide by 4, then complete the square on $x^2+x-\frac{3}{4}=0$.
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What are the solutions of $3x^2-12x+9=0$?
What are the solutions of $3x^2-12x+9=0$?
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$x=1$ or $x=3$. From $(x-2)^2=1$: $x-2=\pm 1$, so $x=2\pm 1$.
$x=1$ or $x=3$. From $(x-2)^2=1$: $x-2=\pm 1$, so $x=2\pm 1$.
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What is the completed-square form of $3x^2-12x+9=0$?
What is the completed-square form of $3x^2-12x+9=0$?
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$\left(x-2\right)^2=1$. First divide by 3, then complete the square on $x^2-4x+3=0$.
$\left(x-2\right)^2=1$. First divide by 3, then complete the square on $x^2-4x+3=0$.
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What are the solutions of $2x^2+8x+6=0$?
What are the solutions of $2x^2+8x+6=0$?
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$x=-1$ or $x=-3$. From $(x+2)^2=1$: $x+2=\pm 1$, so $x=-2\pm 1$.
$x=-1$ or $x=-3$. From $(x+2)^2=1$: $x+2=\pm 1$, so $x=-2\pm 1$.
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What is the completed-square form of $2x^2+8x+6=0$?
What is the completed-square form of $2x^2+8x+6=0$?
Tap to reveal answer
$\left(x+2\right)^2=1$. First divide by 2, then complete the square on $x^2+4x+3=0$.
$\left(x+2\right)^2=1$. First divide by 2, then complete the square on $x^2+4x+3=0$.
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What are the solutions of $x^2-10x+1=0$?
What are the solutions of $x^2-10x+1=0$?
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$x=5\pm 2\sqrt{6}$. From $(x-5)^2=24$: $x=5\pm\sqrt{24}=5\pm 2\sqrt{6}$.
$x=5\pm 2\sqrt{6}$. From $(x-5)^2=24$: $x=5\pm\sqrt{24}=5\pm 2\sqrt{6}$.
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What is the completed-square form of $x^2-10x+1=0$?
What is the completed-square form of $x^2-10x+1=0$?
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$\left(x-5\right)^2=24$. Adding $\left(\frac{-10}{2}\right)^2=25$ to both sides, then simplifying.
$\left(x-5\right)^2=24$. Adding $\left(\frac{-10}{2}\right)^2=25$ to both sides, then simplifying.
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What are the solutions of $x^2+4x-1=0$?
What are the solutions of $x^2+4x-1=0$?
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$x=-2\pm\sqrt{5}$. From $(x+2)^2=5$: $x+2=\pm\sqrt{5}$, so $x=-2\pm\sqrt{5}$.
$x=-2\pm\sqrt{5}$. From $(x+2)^2=5$: $x+2=\pm\sqrt{5}$, so $x=-2\pm\sqrt{5}$.
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What is the completed-square form of $x^2+4x-1=0$?
What is the completed-square form of $x^2+4x-1=0$?
Tap to reveal answer
$\left(x+2\right)^2=5$. Adding $\left(\frac{4}{2}\right)^2=4$ to both sides, then simplifying.
$\left(x+2\right)^2=5$. Adding $\left(\frac{4}{2}\right)^2=4$ to both sides, then simplifying.
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What are the solutions of $x^2-8x+7=0$?
What are the solutions of $x^2-8x+7=0$?
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$x=1$ or $x=7$. From $(x-4)^2=9$: $x-4=\pm 3$, so $x=4\pm 3$.
$x=1$ or $x=7$. From $(x-4)^2=9$: $x-4=\pm 3$, so $x=4\pm 3$.
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What is the completed-square form of $x^2-8x+7=0$?
What is the completed-square form of $x^2-8x+7=0$?
Tap to reveal answer
$\left(x-4\right)^2=9$. Adding $\left(\frac{-8}{2}\right)^2=16$ to both sides, then simplifying.
$\left(x-4\right)^2=9$. Adding $\left(\frac{-8}{2}\right)^2=16$ to both sides, then simplifying.
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What are the solutions of $x^2+6x+5=0$?
What are the solutions of $x^2+6x+5=0$?
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$x=-1$ or $x=-5$. From $(x+3)^2=4$: $x+3=\pm 2$, so $x=-3\pm 2$.
$x=-1$ or $x=-5$. From $(x+3)^2=4$: $x+3=\pm 2$, so $x=-3\pm 2$.
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