Solve by Completing the Square x^2-3/4x-27/64=0

$x^{2} - \frac{3}{4} x - \frac{27}{64} = 0$

Step 1

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

Simplify each term.

Combine $x$ and $\frac{3}{4}$ .

$x^{2} - \frac{x \cdot 3}{4} - \frac{27}{64} = 0$

Move $3$ to the left of $x$ .

$x^{2} - \frac{3 x}{4} - \frac{27}{64} = 0$

Add $\frac{27}{64}$ to both sides of the equation.

$x^{2} - \frac{3 x}{4} = \frac{27}{64}$

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 $b$ .

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

Step 3

Add the term to each side of the equation.

$x^{2} - \frac{3 x}{4} + {(- \frac{3}{8})}^{2} = \frac{27}{64} + {(- \frac{3}{8})}^{2}$

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 $- \frac{3}{8}$ .

$x^{2} - \frac{3 x}{4} + {(- 1)}^{2} {(\frac{3}{8})}^{2} = \frac{27}{64} + {(- \frac{3}{8})}^{2}$

Apply the product rule to $\frac{3}{8}$ .

$x^{2} - \frac{3 x}{4} + {(- 1)}^{2} \frac{3^{2}}{8^{2}} = \frac{27}{64} + {(- \frac{3}{8})}^{2}$

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

$x^{2} - \frac{3 x}{4} + 1 \frac{3^{2}}{8^{2}} = \frac{27}{64} + {(- \frac{3}{8})}^{2}$

Multiply $\frac{3^{2}}{8^{2}}$ by $1$ .

$x^{2} - \frac{3 x}{4} + \frac{3^{2}}{8^{2}} = \frac{27}{64} + {(- \frac{3}{8})}^{2}$

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

$x^{2} - \frac{3 x}{4} + \frac{9}{8^{2}} = \frac{27}{64} + {(- \frac{3}{8})}^{2}$

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

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{27}{64} + {(- \frac{3}{8})}^{2}$

Simplify the right side.

Simplify $\frac{27}{64} + {(- \frac{3}{8})}^{2}$ .

Simplify each term.

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

Apply the product rule to $- \frac{3}{8}$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{27}{64} + {(- 1)}^{2} {(\frac{3}{8})}^{2}$

Apply the product rule to $\frac{3}{8}$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{27}{64} + {(- 1)}^{2} \frac{3^{2}}{8^{2}}$

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

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{27}{64} + 1 \frac{3^{2}}{8^{2}}$

Multiply $\frac{3^{2}}{8^{2}}$ by $1$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{27}{64} + \frac{3^{2}}{8^{2}}$

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

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{27}{64} + \frac{9}{8^{2}}$

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

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{27}{64} + \frac{9}{64}$

Simplify terms.

Combine the numerators over the common denominator.

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{27 + 9}{64}$

Add $27$ and $9$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{36}{64}$

Cancel the common factor of $36$ and $64$ .

Factor $4$ out of $36$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{4 (9)}{64}$

Cancel the common factors.

Factor $4$ out of $64$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{4 \cdot 9}{4 \cdot 16}$

Cancel the common factor.

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{4 \cdot 9}{4 \cdot 16}$

Rewrite the expression.

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{9}{16}$

Step 5

Factor the perfect trinomial square into ${(x - \frac{3}{8})}^{2}$ .

${(x - \frac{3}{8})}^{2} = \frac{9}{16}$

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.

$x - \frac{3}{8} = \pm \sqrt{\frac{9}{16}}$

Simplify $\pm \sqrt{\frac{9}{16}}$ .

Rewrite $\sqrt{\frac{9}{16}}$ as $\frac{\sqrt{9}}{\sqrt{16}}$ .

$x - \frac{3}{8} = \pm \frac{\sqrt{9}}{\sqrt{16}}$

Simplify the numerator.

Rewrite $9$ as $3^{2}$ .

$x - \frac{3}{8} = \pm \frac{\sqrt{3^{2}}}{\sqrt{16}}$

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

$x - \frac{3}{8} = \pm \frac{3}{\sqrt{16}}$

Simplify the denominator.

Rewrite $16$ as $4^{2}$ .

$x - \frac{3}{8} = \pm \frac{3}{\sqrt{4^{2}}}$

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

$x - \frac{3}{8} = \pm \frac{3}{4}$

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.

$x - \frac{3}{8} = \frac{3}{4}$

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

Add $\frac{3}{8}$ to both sides of the equation.

$x = \frac{3}{4} + \frac{3}{8}$

To write $\frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{3}{4} \cdot \frac{2}{2} + \frac{3}{8}$

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

Multiply $\frac{3}{4}$ by $\frac{2}{2}$ .

$x = \frac{3 \cdot 2}{4 \cdot 2} + \frac{3}{8}$

Multiply $4$ by $2$ .

$x = \frac{3 \cdot 2}{8} + \frac{3}{8}$

Combine the numerators over the common denominator.

$x = \frac{3 \cdot 2 + 3}{8}$

Simplify the numerator.

Multiply $3$ by $2$ .

$x = \frac{6 + 3}{8}$

Add $6$ and $3$ .

$x = \frac{9}{8}$

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

$x - \frac{3}{8} = - \frac{3}{4}$

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

Add $\frac{3}{8}$ to both sides of the equation.

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

To write $- \frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

Multiply $\frac{3}{4}$ by $\frac{2}{2}$ .

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

Multiply $4$ by $2$ .

$x = - \frac{3 \cdot 2}{8} + \frac{3}{8}$

Combine the numerators over the common denominator.

$x = \frac{- 3 \cdot 2 + 3}{8}$

Simplify the numerator.

Multiply $- 3$ by $2$ .

$x = \frac{- 6 + 3}{8}$

Add $- 6$ and $3$ .

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

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.

$x = \frac{9}{8}, - \frac{3}{8}$

Solve by Completing the Square x^2-3/4x-27/64=0

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