Solve by Completing the Square 8x^2-8x-1=0

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

Step 1

Add $1$ to both sides of the equation.

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

Step 2

Divide each term in $8 x^{2} - 8 x = 1$ by $8$ and simplify.

Divide each term in $8 x^{2} - 8 x = 1$ by $8$ .

$\frac{8 x^{2}}{8} + \frac{- 8 x}{8} = \frac{1}{8}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x^{2}}{8} + \frac{- 8 x}{8} = \frac{1}{8}$

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{- 8 x}{8} = \frac{1}{8}$

Cancel the common factor of $- 8$ and $8$ .

Factor $8$ out of $- 8 x$ .

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

Cancel the common factors.

Factor $8$ out of $8$ .

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

Cancel the common factor.

$x^{2} + \frac{8 (- x)}{8 \cdot 1} = \frac{1}{8}$

Rewrite the expression.

$x^{2} + \frac{- x}{1} = \frac{1}{8}$

Divide $- x$ by $1$ .

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

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 $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{1}{2})}^{2}$

Step 4

Add the term to each side of the equation.

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

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{1}{2}$ .

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

Apply the product rule to $\frac{1}{2}$ .

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

Raise $- 1$ to the power of $2$ .

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

Multiply $\frac{1^{2}}{2^{2}}$ by $1$ .

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

One to any power is one.

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

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

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

Simplify the right side.

Simplify $\frac{1}{8} + {(- \frac{1}{2})}^{2}$ .

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{1}{2}$ .

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

Apply the product rule to $\frac{1}{2}$ .

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

Raise $- 1$ to the power of $2$ .

$x^{2} - x + \frac{1}{4} = \frac{1}{8} + 1 \frac{1^{2}}{2^{2}}$

Multiply $\frac{1^{2}}{2^{2}}$ by $1$ .

$x^{2} - x + \frac{1}{4} = \frac{1}{8} + \frac{1^{2}}{2^{2}}$

One to any power is one.

$x^{2} - x + \frac{1}{4} = \frac{1}{8} + \frac{1}{2^{2}}$

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

$x^{2} - x + \frac{1}{4} = \frac{1}{8} + \frac{1}{4}$

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{2} - x + \frac{1}{4} = \frac{1}{8} + \frac{1}{4} \cdot \frac{2}{2}$

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{4}$ by $\frac{2}{2}$ .

$x^{2} - x + \frac{1}{4} = \frac{1}{8} + \frac{2}{4 \cdot 2}$

Multiply $4$ by $2$ .

$x^{2} - x + \frac{1}{4} = \frac{1}{8} + \frac{2}{8}$

Combine the numerators over the common denominator.

$x^{2} - x + \frac{1}{4} = \frac{1 + 2}{8}$

Add $1$ and $2$ .

$x^{2} - x + \frac{1}{4} = \frac{3}{8}$

Step 6

Factor the perfect trinomial square into ${(x - \frac{1}{2})}^{2}$ .

${(x - \frac{1}{2})}^{2} = \frac{3}{8}$

Step 7

Solve the equation for $x$ .

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

$x - \frac{1}{2} = \pm \sqrt{\frac{3}{8}}$

Simplify $\pm \sqrt{\frac{3}{8}}$ .

Rewrite $\sqrt{\frac{3}{8}}$ as $\frac{\sqrt{3}}{\sqrt{8}}$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{3}}{\sqrt{8}}$

Simplify the denominator.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{3}}{\sqrt{4 (2)}}$

Rewrite $4$ as $2^{2}$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{3}}{\sqrt{2^{2} \cdot 2}}$

Pull terms out from under the radical.

$x - \frac{1}{2} = \pm \frac{\sqrt{3}}{2 \sqrt{2}}$

Multiply $\frac{\sqrt{3}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{3}}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{3} \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Move $\sqrt{2}$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{3} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{3} \sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{3} \sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{3} \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x - \frac{1}{2} = \pm \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Rewrite the expression.

$x - \frac{1}{2} = \pm \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2^{1}}$

Evaluate the exponent.

$x - \frac{1}{2} = \pm \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}$

Simplify the numerator.

Combine using the product rule for radicals.

$x - \frac{1}{2} = \pm \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}$

Multiply $3$ by $2$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{6}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{6}}{4}$

Add $\frac{1}{2}$ to both sides of the equation.

$x = \pm \frac{\sqrt{6}}{4} + \frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{6}}{4} + \frac{1}{2}$

Decimal Form:

$x = 1.11237243 \dots, - 0.11237243 \dots$

Solve by Completing the Square 8x^2-8x-1=0

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