Solve by Completing the Square 8x^2+6x=9

$8 x^{2} + 6 x = 9$

Step 1

Divide each term in $8 x^{2} + 6 x = 9$ by $8$ and simplify.

Divide each term in $8 x^{2} + 6 x = 9$ by $8$ .

$\frac{8 x^{2}}{8} + \frac{6 x}{8} = \frac{9}{8}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x^{2}}{8} + \frac{6 x}{8} = \frac{9}{8}$

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

$x^{2} + \frac{6 x}{8} = \frac{9}{8}$

Cancel the common factor of $6$ and $8$ .

Factor $2$ out of $6 x$ .

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

Cancel the common factors.

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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.

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

Step 4

Simplify the equation.

Simplify the left side.

Simplify each term.

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

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

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

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

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

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

Simplify the right side.

Simplify $\frac{9}{8} + {(\frac{3}{8})}^{2}$ .

Simplify each term.

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

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

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

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

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

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

To write $\frac{9}{8}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

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

Write each expression with a common denominator of $64$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{9}{8}$ by $\frac{8}{8}$ .

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

Multiply $8$ by $8$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $9$ by $8$ .

$x^{2} + \frac{3 x}{4} + \frac{9}{64} = \frac{72 + 9}{64}$

Add $72$ and $9$ .

$x^{2} + \frac{3 x}{4} + \frac{9}{64} = \frac{81}{64}$

Step 5

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

${(x + \frac{3}{8})}^{2} = \frac{81}{64}$

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.

$x + \frac{3}{8} = \pm \sqrt{\frac{81}{64}}$

Simplify $\pm \sqrt{\frac{81}{64}}$ .

Rewrite $\sqrt{\frac{81}{64}}$ as $\frac{\sqrt{81}}{\sqrt{64}}$ .

$x + \frac{3}{8} = \pm \frac{\sqrt{81}}{\sqrt{64}}$

Simplify the numerator.

Rewrite $81$ as $9^{2}$ .

$x + \frac{3}{8} = \pm \frac{\sqrt{9^{2}}}{\sqrt{64}}$

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

$x + \frac{3}{8} = \pm \frac{9}{\sqrt{64}}$

Simplify the denominator.

Rewrite $64$ as $8^{2}$ .

$x + \frac{3}{8} = \pm \frac{9}{\sqrt{8^{2}}}$

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

$x + \frac{3}{8} = \pm \frac{9}{8}$

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.

$x + \frac{3}{8} = \frac{9}{8}$

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

Subtract $\frac{3}{8}$ from both sides of the equation.

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

Combine the numerators over the common denominator.

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

Subtract $3$ from $9$ .

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

Cancel the common factor of $6$ and $8$ .

Factor $2$ out of $6$ .

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

Cancel the common factors.

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Subtract $\frac{3}{8}$ from both sides of the equation.

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

Combine the numerators over the common denominator.

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

Subtract $3$ from $- 9$ .

$x = \frac{- 12}{8}$

Cancel the common factor of $- 12$ and $8$ .

Factor $4$ out of $- 12$ .

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

Cancel the common factors.

Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

Solve by Completing the Square 8x^2+6x=9

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