Solve by Completing the Square x^2-1/4x-15/64=0

$x^{2} - \frac{1}{4} x - \frac{15}{64} = 0$

Step 1

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

Combine $x$ and $\frac{1}{4}$ .

$x^{2} - \frac{x}{4} - \frac{15}{64} = 0$

Add $\frac{15}{64}$ to both sides of the equation.

$x^{2} - \frac{x}{4} = \frac{15}{64}$

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 $b$ .

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

Step 3

Add the term to each side of the equation.

$x^{2} - \frac{x}{4} + {(- \frac{1}{8})}^{2} = \frac{15}{64} + {(- \frac{1}{8})}^{2}$

Step 4

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 $- \frac{1}{8}$ .

$x^{2} - \frac{x}{4} + {(- 1)}^{2} {(\frac{1}{8})}^{2} = \frac{15}{64} + {(- \frac{1}{8})}^{2}$

Apply the product rule to $\frac{1}{8}$ .

$x^{2} - \frac{x}{4} + {(- 1)}^{2} \frac{1^{2}}{8^{2}} = \frac{15}{64} + {(- \frac{1}{8})}^{2}$

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

$x^{2} - \frac{x}{4} + 1 \frac{1^{2}}{8^{2}} = \frac{15}{64} + {(- \frac{1}{8})}^{2}$

Multiply $\frac{1^{2}}{8^{2}}$ by $1$ .

$x^{2} - \frac{x}{4} + \frac{1^{2}}{8^{2}} = \frac{15}{64} + {(- \frac{1}{8})}^{2}$

One to any power is one.

$x^{2} - \frac{x}{4} + \frac{1}{8^{2}} = \frac{15}{64} + {(- \frac{1}{8})}^{2}$

Raise $8$ to the power of $2$ .

$x^{2} - \frac{x}{4} + \frac{1}{64} = \frac{15}{64} + {(- \frac{1}{8})}^{2}$

Simplify the right side.

Simplify $\frac{15}{64} + {(- \frac{1}{8})}^{2}$ .

Simplify each term.

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

Apply the product rule to $- \frac{1}{8}$ .

$x^{2} - \frac{x}{4} + \frac{1}{64} = \frac{15}{64} + {(- 1)}^{2} {(\frac{1}{8})}^{2}$

Apply the product rule to $\frac{1}{8}$ .

$x^{2} - \frac{x}{4} + \frac{1}{64} = \frac{15}{64} + {(- 1)}^{2} \frac{1^{2}}{8^{2}}$

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

$x^{2} - \frac{x}{4} + \frac{1}{64} = \frac{15}{64} + 1 \frac{1^{2}}{8^{2}}$

Multiply $\frac{1^{2}}{8^{2}}$ by $1$ .

$x^{2} - \frac{x}{4} + \frac{1}{64} = \frac{15}{64} + \frac{1^{2}}{8^{2}}$

One to any power is one.

$x^{2} - \frac{x}{4} + \frac{1}{64} = \frac{15}{64} + \frac{1}{8^{2}}$

Raise $8$ to the power of $2$ .

$x^{2} - \frac{x}{4} + \frac{1}{64} = \frac{15}{64} + \frac{1}{64}$

Simplify terms.

Combine the numerators over the common denominator.

$x^{2} - \frac{x}{4} + \frac{1}{64} = \frac{15 + 1}{64}$

Add $15$ and $1$ .

$x^{2} - \frac{x}{4} + \frac{1}{64} = \frac{16}{64}$

Cancel the common factor of $16$ and $64$ .

Factor $16$ out of $16$ .

$x^{2} - \frac{x}{4} + \frac{1}{64} = \frac{16 (1)}{64}$

Cancel the common factors.

Factor $16$ out of $64$ .

$x^{2} - \frac{x}{4} + \frac{1}{64} = \frac{16 \cdot 1}{16 \cdot 4}$

Cancel the common factor.

$x^{2} - \frac{x}{4} + \frac{1}{64} = \frac{16 \cdot 1}{16 \cdot 4}$

Rewrite the expression.

$x^{2} - \frac{x}{4} + \frac{1}{64} = \frac{1}{4}$

Step 5

Factor the perfect trinomial square into ${(x - \frac{1}{8})}^{2}$ .

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

Step 6

Solve the equation for $x$ .

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

$x - \frac{1}{8} = \pm \sqrt{\frac{1}{4}}$

Simplify $\pm \sqrt{\frac{1}{4}}$ .

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$x - \frac{1}{8} = \pm \frac{\sqrt{1}}{\sqrt{4}}$

Any root of $1$ is $1$ .

$x - \frac{1}{8} = \pm \frac{1}{\sqrt{4}}$

Simplify the denominator.

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

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

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

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

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

First, use the positive value of the $\pm$ to find the first solution.

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

Move all terms not containing $x$ to the right side of the equation.

Add $\frac{1}{8}$ to both sides of the equation.

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

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

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

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

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

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

Multiply $2$ by $4$ .

$x = \frac{4}{8} + \frac{1}{8}$

Combine the numerators over the common denominator.

$x = \frac{4 + 1}{8}$

Add $4$ and $1$ .

$x = \frac{5}{8}$

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

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

Move all terms not containing $x$ to the right side of the equation.

Add $\frac{1}{8}$ to both sides of the equation.

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

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

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

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

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

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

Multiply $2$ by $4$ .

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

Combine the numerators over the common denominator.

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

Add $- 4$ and $1$ .

$x = \frac{- 3}{8}$

Move the negative in front of the fraction.

$x = - \frac{3}{8}$

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

$x = \frac{5}{8}, - \frac{3}{8}$

Solve by Completing the Square x^2-1/4x-15/64=0

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