Examples

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

Step 1

Plug in $0$ for $y$ .

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

Step 2

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

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Step 2.1

Remove parentheses.

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

Step 2.2

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

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

Step 2.3

Subtract $16$ from both sides of the equation.

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

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 $b$ .

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

Step 4

Add the term to each side of the equation.

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

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 $- 3$ to the power of $2$ .

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

Step 5.2

Simplify the right side.

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Step 5.2.1

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

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Step 5.2.1.1

Raise $- 3$ to the power of $2$ .

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

Step 5.2.1.2

Add $- 16$ and $9$ .

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

Step 6

Factor the perfect trinomial square into ${(x - 3)}^{2}$ .

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

Step 7

Solve the equation for $x$ .

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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.

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

Step 7.2

Simplify $\pm \sqrt{- 7}$ .

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Step 7.2.1

Rewrite $- 7$ as $- 1 (7)$ .

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

Step 7.2.2

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

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

Step 7.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

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

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 $\pm$ to find the first solution.

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

Step 7.3.2

Add $3$ to both sides of the equation.

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

Step 7.3.3

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 7.3.4

Add $3$ to both sides of the equation.

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

Step 7.3.5

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

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

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