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High School Math : Completing the Square

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Example Questions

Example Question #1 : Completing The Square

$f(x)=x^{2}-4x+7$

Find the vertex of the parabola by completing the square.

Possible Answers:

$\small (7,0)$

$\small (2,3)$

$\small (1,4)$

$\small (3,2)$

$\small (0,7)$

Correct answer:

$\small (2,3)$

Explanation:

To find the vertex of a parabola, we must put the equation into the vertex form:

$\small f(x)=a[b(x-h)]^{2}+k$

The vertex can then be found with the coordinates (h, k).

To put the parabola's equation into vertex form, you have to complete the square. Completing the square just means adding the same number to both sides of the equation -- which, remember, doesn't change the value of the equation -- in order to create a perfect square.

Start with the original equation:

$f(x)=x^{2}-4x+7$

Put all of the terms on one side:

$f(x)-7=x^{2}-4x$

Now we know that we have to add something to both sides in order to create a perfect square:

$f(x)+7+\square =x^{2}-4x+\square$

In this case, we need to add 4 on both sides so that the right-hand side of the equation factors neatly.

$f(x)-7+4=x^{2}-4x+4$

Now we factor:

$f(x)-3=(x-2)^{2}$

Once we isolate f(x) , we have the equation in vertex form:

$f(x)=(x-2)^{2}+3$

Thus, the parabola's vertex can be found at $\small (2,3)$ .

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Example Question #2 : Completing The Square

Complete the square:

2x^2+3x+2=0

Possible Answers:

$x=\frac{-2 \pm i\sqrt{7}}{4}$

$x=\frac{-2 \pm i\sqrt{7}}{3}$

$x=\frac{-3 \pm i\sqrt{7}}{4}$

$x=\frac{-1 \pm i\sqrt{7}}{4}$

$x=\frac{-3 \pm i\sqrt{7}}{3}$

Correct answer:

$x=\frac{-3 \pm i\sqrt{7}}{4}$

Explanation:

Begin by dividing the equation by and subtracting from each side:

2x^2+3x+2=0

$x^2+\frac{3}{2}x = -1$

Square the value in front of the and add to each side:

$x^2+\frac{3}{2}x +\frac{9}{16} = -1+\frac{9}{16}$

Factor the left side of the equation:

$(x+\frac{3}{4})^2 = \frac{-7}{16}$

Take the square root of both sides and simplify:

$x+\frac{3}{4} = \frac{\pm \sqrt{-7}}{4}$

$x = \frac{-3 \pm i\sqrt{7}}{4}$

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Example Question #3 : Completing The Square

Use factoring to solve the quadratic equation:

6x^2+7x-3=0

Possible Answers:

$x=-\frac{3}{2},\frac{1}{3}$

$x=-\frac{3}{2},\frac{1}{4}$

$x=-\frac{3}{4},\frac{1}{3}$

$x=-\frac{3}{4},\frac{1}{4}$

$x=-\frac{3}{2},\frac{2}{3}$

Correct answer:

$x=-\frac{3}{2},\frac{1}{3}$

Explanation:

Factor and solve:

6x^2+7x-3=0

6x^2 +9x-2x-3=0

Factor like terms:

3x(2x+3)-1(2x+3)=0

Combine like terms:

(2x+3)(3x-1)=0

$x=-\frac{3}{2},\frac{1}{3}$

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Example Question #4 : Completing The Square

Complete the square:

2n^2+n-21=0

Possible Answers:

$n=-\frac{5}{2},3$

$n=-\frac{7}{2},3$

$n=-\frac{7}{2},4$

$n=-\frac{9}{2},3$

$n=-\frac{7}{2},2$

Correct answer:

$n=-\frac{7}{2},3$

Explanation:

Begin by dividing the equation by and adding 10.5 to each side:

2n^2+n-21=0

$n^2+\frac{1}{2}n = \frac{21}{2}$

Square the value in front of the and add to each side:

$n^2+\frac{1}{2}n + \frac{1}{16} = \frac{21}{2} + \frac{1}{16}$

Factor the left side of the equation:

$(n+\frac{1}{4})^2 = \frac{169}{16}$

Take the square root of both sides and simplify:

$n+\frac{1}{4} = \frac{\pm 13}{4}$

$n = \frac{-1 \pm 13}{4}$

$n=-\frac{7}{2},3$

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Example Question #5 : Completing The Square

Complete the square:

3x^2+4x+2=0

Possible Answers:

$x=\frac{-4 \pm i\sqrt{2}}{3}$

$x=\frac{-3 \pm i\sqrt{2}}{3}$

$x=\frac{-2 \pm i\sqrt{2}}{4}$

$x=\frac{-2 \pm i\sqrt{3}}{3}$

$x=\frac{-2 \pm i\sqrt{2}}{3}$

Correct answer:

$x=\frac{-2 \pm i\sqrt{2}}{3}$

Explanation:

Begin by dividing the equation by and subtracting $\frac{2}{3}$ from each side:

3x^2+4x+2=0

$x^2+\frac{4}{3}x = -\frac{2}{3}$

Square the value in front of the and add to each side:

$x^2+\frac{4}{3}x +\frac{4}{9} = -\frac{2}{3}+\frac{4}{9}$

Factor the left side of the equation:

$(x+\frac{2}{3})^2 = \frac{-2}{9}$

Take the square root of both sides and simplify:

$x+\frac{2}{3} = \frac{\pm \sqrt{-2}}{3}$

$x = \frac{-2 \pm i\sqrt{2}}{3}$

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High School Math : Completing the Square

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Completing The Square

Example Question #2 : Completing The Square

Example Question #3 : Completing The Square

Example Question #4 : Completing The Square

Example Question #5 : Completing The Square

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