Solve by Completing the Square x^2-3/2x-27/16=0

$x^{2} - \frac{3}{2} x - \frac{27}{16} = 0$

Step 1

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

Simplify each term.

Combine $x$ and $\frac{3}{2}$ .

$x^{2} - \frac{x \cdot 3}{2} - \frac{27}{16} = 0$

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

$x^{2} - \frac{3 x}{2} - \frac{27}{16} = 0$

Add $\frac{27}{16}$ to both sides of the equation.

$x^{2} - \frac{3 x}{2} = \frac{27}{16}$

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 $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{3}{4})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - \frac{3 x}{2} + {(- \frac{3}{4})}^{2} = \frac{27}{16} + {(- \frac{3}{4})}^{2}$

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 $- \frac{3}{4}$ .

$x^{2} - \frac{3 x}{2} + {(- 1)}^{2} {(\frac{3}{4})}^{2} = \frac{27}{16} + {(- \frac{3}{4})}^{2}$

Apply the product rule to $\frac{3}{4}$ .

$x^{2} - \frac{3 x}{2} + {(- 1)}^{2} \frac{3^{2}}{4^{2}} = \frac{27}{16} + {(- \frac{3}{4})}^{2}$

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

$x^{2} - \frac{3 x}{2} + 1 \frac{3^{2}}{4^{2}} = \frac{27}{16} + {(- \frac{3}{4})}^{2}$

Multiply $\frac{3^{2}}{4^{2}}$ by $1$ .

$x^{2} - \frac{3 x}{2} + \frac{3^{2}}{4^{2}} = \frac{27}{16} + {(- \frac{3}{4})}^{2}$

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

$x^{2} - \frac{3 x}{2} + \frac{9}{4^{2}} = \frac{27}{16} + {(- \frac{3}{4})}^{2}$

Raise $4$ to the power of $2$ .

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{27}{16} + {(- \frac{3}{4})}^{2}$

Simplify the right side.

Simplify $\frac{27}{16} + {(- \frac{3}{4})}^{2}$ .

Simplify each term.

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

Apply the product rule to $- \frac{3}{4}$ .

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{27}{16} + {(- 1)}^{2} {(\frac{3}{4})}^{2}$

Apply the product rule to $\frac{3}{4}$ .

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{27}{16} + {(- 1)}^{2} \frac{3^{2}}{4^{2}}$

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

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{27}{16} + 1 \frac{3^{2}}{4^{2}}$

Multiply $\frac{3^{2}}{4^{2}}$ by $1$ .

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{27}{16} + \frac{3^{2}}{4^{2}}$

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

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{27}{16} + \frac{9}{4^{2}}$

Raise $4$ to the power of $2$ .

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{27}{16} + \frac{9}{16}$

Simplify terms.

Combine the numerators over the common denominator.

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{27 + 9}{16}$

Add $27$ and $9$ .

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{36}{16}$

Cancel the common factor of $36$ and $16$ .

Factor $4$ out of $36$ .

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

Cancel the common factors.

Factor $4$ out of $16$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Factor the perfect trinomial square into ${(x - \frac{3}{4})}^{2}$ .

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

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.

$x - \frac{3}{4} = \pm \sqrt{\frac{9}{4}}$

Simplify $\pm \sqrt{\frac{9}{4}}$ .

Rewrite $\sqrt{\frac{9}{4}}$ as $\frac{\sqrt{9}}{\sqrt{4}}$ .

$x - \frac{3}{4} = \pm \frac{\sqrt{9}}{\sqrt{4}}$

Simplify the numerator.

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

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

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

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

Simplify the denominator.

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

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

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

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

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.

$x - \frac{3}{4} = \frac{3}{2}$

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

Add $\frac{3}{4}$ to both sides of the equation.

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

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

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

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $2$ .

$x = \frac{6 + 3}{4}$

Add $6$ and $3$ .

$x = \frac{9}{4}$

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

$x - \frac{3}{4} = - \frac{3}{2}$

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

Add $\frac{3}{4}$ to both sides of the equation.

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

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

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

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 3$ by $2$ .

$x = \frac{- 6 + 3}{4}$

Add $- 6$ and $3$ .

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

Move the negative in front of the fraction.

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

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

$x = \frac{9}{4}, - \frac{3}{4}$

Solve by Completing the Square x^2-3/2x-27/16=0

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