Examples

y = x^{2} - 6 x + 16

$y = x^{2} - 6 x + 16$

Step 1

Plug in

0

$0$ for

y

$y$ .

0 = x^{2} - 6 x + 16

$0 = x^{2} - 6 x + 16$

Step 2

Simplify the equation into a proper form to complete the square.

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Step 2.1

Remove parentheses.

0 = x^{2} - 6 x + 16

$0 = x^{2} - 6 x + 16$

Step 2.2

Since

x

$x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

x^{2} - 6 x + 16 = 0

$x^{2} - 6 x + 16 = 0$

Step 2.3

Subtract

16

$16$ from both sides of the equation.

x^{2} - 6 x = - 16

$x^{2} - 6 x = - 16$

x^{2} - 6 x = - 16

$x^{2} - 6 x = - 16$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of

b

$b$ .

{(\frac{b}{2})}^{2} = {(- 3)}^{2}

${(\frac{b}{2})}^{2} = {(- 3)}^{2}$

Step 4

Add the term to each side of the equation.

x^{2} - 6 x + {(- 3)}^{2} = - 16 + {(- 3)}^{2}

$x^{2} - 6 x + {(- 3)}^{2} = - 16 + {(- 3)}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Raise

- 3

$- 3$ to the power of

2

$2$ .

x^{2} - 6 x + 9 = - 16 + {(- 3)}^{2}

$x^{2} - 6 x + 9 = - 16 + {(- 3)}^{2}$

x^{2} - 6 x + 9 = - 16 + {(- 3)}^{2}

$x^{2} - 6 x + 9 = - 16 + {(- 3)}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify

- 16 + {(- 3)}^{2}

$- 16 + {(- 3)}^{2}$ .

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Step 5.2.1.1

Raise

- 3

$- 3$ to the power of

2

$2$ .

x^{2} - 6 x + 9 = - 16 + 9

$x^{2} - 6 x + 9 = - 16 + 9$

Step 5.2.1.2

Add

- 16

$- 16$ and

9

$9$ .

x^{2} - 6 x + 9 = - 7

$x^{2} - 6 x + 9 = - 7$

x^{2} - 6 x + 9 = - 7

$x^{2} - 6 x + 9 = - 7$

x^{2} - 6 x + 9 = - 7

$x^{2} - 6 x + 9 = - 7$

x^{2} - 6 x + 9 = - 7

$x^{2} - 6 x + 9 = - 7$

Step 6

Factor the perfect trinomial square into

{(x - 3)}^{2}

${(x - 3)}^{2}$ .

{(x - 3)}^{2} = - 7

${(x - 3)}^{2} = - 7$

Step 7

Solve the equation for

x

$x$ .

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Step 7.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

x - 3 = \pm \sqrt{- 7}

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

Step 7.2

Simplify

\pm \sqrt{- 7}

$\pm \sqrt{- 7}$ .

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Step 7.2.1

Rewrite

- 7

$- 7$ as

- 1 (7)

$- 1 (7)$ .

x - 3 = \pm \sqrt{- 1 (7)}

$x - 3 = \pm \sqrt{- 1 (7)}$

Step 7.2.2

Rewrite

\sqrt{- 1 (7)}

$\sqrt{- 1 (7)}$ as

\sqrt{- 1} \cdot \sqrt{7}

$\sqrt{- 1} \cdot \sqrt{7}$ .

x - 3 = \pm \sqrt{- 1} \cdot \sqrt{7}

$x - 3 = \pm \sqrt{- 1} \cdot \sqrt{7}$

Step 7.2.3

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

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

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

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

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

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.3.1

First, use the positive value of the

\pm

$\pm$ to find the first solution.

x - 3 = i \sqrt{7}

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

Step 7.3.2

Add

3

$3$ to both sides of the equation.

x = i \sqrt{7} + 3

$x = i \sqrt{7} + 3$

Step 7.3.3

Next, use the negative value of the

\pm

$\pm$ to find the second solution.

x - 3 = - i \sqrt{7}

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

Step 7.3.4

Add

3

$3$ to both sides of the equation.

x = - i \sqrt{7} + 3

$x = - i \sqrt{7} + 3$

Step 7.3.5

The complete solution is the result of both the positive and negative portions of the solution.

x = i \sqrt{7} + 3, - i \sqrt{7} + 3

$x = i \sqrt{7} + 3, - i \sqrt{7} + 3$

x = i \sqrt{7} + 3, - i \sqrt{7} + 3

$x = i \sqrt{7} + 3, - i \sqrt{7} + 3$

x = i \sqrt{7} + 3, - i \sqrt{7} + 3

$x = i \sqrt{7} + 3, - i \sqrt{7} + 3$

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