Examples

x^{2} + 10 x + 16 = 0

$x^{2} + 10 x + 16 = 0$

Step 1

Subtract

16

$16$ from both sides of the equation.

x^{2} + 10 x = - 16

$x^{2} + 10 x = - 16$

Step 2

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} = {(5)}^{2}

${(\frac{b}{2})}^{2} = {(5)}^{2}$

Step 3

Add the term to each side of the equation.

x^{2} + 10 x + {(5)}^{2} = - 16 + {(5)}^{2}

$x^{2} + 10 x + {(5)}^{2} = - 16 + {(5)}^{2}$

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Raise

5

$5$ to the power of

2

$2$ .

x^{2} + 10 x + 25 = - 16 + {(5)}^{2}

$x^{2} + 10 x + 25 = - 16 + {(5)}^{2}$

x^{2} + 10 x + 25 = - 16 + {(5)}^{2}

$x^{2} + 10 x + 25 = - 16 + {(5)}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify

- 16 + {(5)}^{2}

$- 16 + {(5)}^{2}$ .

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Step 4.2.1.1

Raise

5

$5$ to the power of

2

$2$ .

x^{2} + 10 x + 25 = - 16 + 25

$x^{2} + 10 x + 25 = - 16 + 25$

Step 4.2.1.2

Add

- 16

$- 16$ and

25

$25$ .

x^{2} + 10 x + 25 = 9

$x^{2} + 10 x + 25 = 9$

x^{2} + 10 x + 25 = 9

$x^{2} + 10 x + 25 = 9$

x^{2} + 10 x + 25 = 9

$x^{2} + 10 x + 25 = 9$

x^{2} + 10 x + 25 = 9

$x^{2} + 10 x + 25 = 9$

Step 5

Factor the perfect trinomial square into

{(x + 5)}^{2}

${(x + 5)}^{2}$ .

{(x + 5)}^{2} = 9

${(x + 5)}^{2} = 9$

Step 6

Solve the equation for

x

$x$ .

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Step 6.1

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

x + 5 = \pm \sqrt{9}

$x + 5 = \pm \sqrt{9}$

Step 6.2

Simplify

\pm \sqrt{9}

$\pm \sqrt{9}$ .

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Step 6.2.1

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

x + 5 = \pm \sqrt{3^{2}}

$x + 5 = \pm \sqrt{3^{2}}$

Step 6.2.2

Pull terms out from under the radical, assuming positive real numbers.

x + 5 = \pm 3

$x + 5 = \pm 3$

x + 5 = \pm 3

$x + 5 = \pm 3$

Step 6.3

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

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Step 6.3.1

First, use the positive value of the

\pm

$\pm$ to find the first solution.

x + 5 = 3

$x + 5 = 3$

Step 6.3.2

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 6.3.2.1

Subtract

5

$5$ from both sides of the equation.

x = 3 - 5

$x = 3 - 5$

Step 6.3.2.2

Subtract

5

$5$ from

3

$3$ .

x = - 2

$x = - 2$

x = - 2

$x = - 2$

Step 6.3.3

Next, use the negative value of the

\pm

$\pm$ to find the second solution.

x + 5 = - 3

$x + 5 = - 3$

Step 6.3.4

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 6.3.4.1

Subtract

5

$5$ from both sides of the equation.

x = - 3 - 5

$x = - 3 - 5$

Step 6.3.4.2

Subtract

5

$5$ from

- 3

$- 3$ .

x = - 8

$x = - 8$

x = - 8

$x = - 8$

Step 6.3.5

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

x = - 2, - 8

$x = - 2, - 8$

x = - 2, - 8

$x = - 2, - 8$

x = - 2, - 8

$x = - 2, - 8$

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