Examples

x^{2} - 8 x - 1 = 0

$x^{2} - 8 x - 1 = 0$

Step 1

Add

1

$1$ to both sides of the equation.

x^{2} - 8 x = 1

$x^{2} - 8 x = 1$

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} = {(- 4)}^{2}

${(\frac{b}{2})}^{2} = {(- 4)}^{2}$

Step 3

Add the term to each side of the equation.

x^{2} - 8 x + {(- 4)}^{2} = 1 + {(- 4)}^{2}

$x^{2} - 8 x + {(- 4)}^{2} = 1 + {(- 4)}^{2}$

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

- 4

$- 4$ to the power of

2

$2$ .

x^{2} - 8 x + 16 = 1 + {(- 4)}^{2}

$x^{2} - 8 x + 16 = 1 + {(- 4)}^{2}$

x^{2} - 8 x + 16 = 1 + {(- 4)}^{2}

$x^{2} - 8 x + 16 = 1 + {(- 4)}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify

1 + {(- 4)}^{2}

$1 + {(- 4)}^{2}$ .

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Step 4.2.1.1

Raise

- 4

$- 4$ to the power of

2

$2$ .

x^{2} - 8 x + 16 = 1 + 16

$x^{2} - 8 x + 16 = 1 + 16$

Step 4.2.1.2

Add

1

$1$ and

16

$16$ .

x^{2} - 8 x + 16 = 17

$x^{2} - 8 x + 16 = 17$

x^{2} - 8 x + 16 = 17

$x^{2} - 8 x + 16 = 17$

x^{2} - 8 x + 16 = 17

$x^{2} - 8 x + 16 = 17$

x^{2} - 8 x + 16 = 17

$x^{2} - 8 x + 16 = 17$

Step 5

Factor the perfect trinomial square into

{(x - 4)}^{2}

${(x - 4)}^{2}$ .

{(x - 4)}^{2} = 17

${(x - 4)}^{2} = 17$

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 - 4 = \pm \sqrt{17}

$x - 4 = \pm \sqrt{17}$

Step 6.2

Add

4

$4$ to both sides of the equation.

x = \pm \sqrt{17} + 4

$x = \pm \sqrt{17} + 4$

x = \pm \sqrt{17} + 4

$x = \pm \sqrt{17} + 4$

Step 7

The result can be shown in multiple forms.

Exact Form:

x = \pm \sqrt{17} + 4

$x = \pm \sqrt{17} + 4$

Decimal Form:

x = 8.12310562 \dots, - 0.12310562 \dots

$x = 8.12310562 \dots, - 0.12310562 \dots$

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