Solve by Completing the Square x^2+1/2x+1/16=4/9

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

Step 1

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

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

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

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

Subtract $\frac{1}{16}$ from both sides of the equation.

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

To write $\frac{4}{9}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

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

To write $- \frac{1}{16}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

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

Write each expression with a common denominator of $144$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4}{9}$ by $\frac{16}{16}$ .

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

Multiply $9$ by $16$ .

$x^{2} + \frac{x}{2} = \frac{4 \cdot 16}{144} - \frac{1}{16} \cdot \frac{9}{9}$

Multiply $\frac{1}{16}$ by $\frac{9}{9}$ .

$x^{2} + \frac{x}{2} = \frac{4 \cdot 16}{144} - \frac{9}{16 \cdot 9}$

Multiply $16$ by $9$ .

$x^{2} + \frac{x}{2} = \frac{4 \cdot 16}{144} - \frac{9}{144}$

Combine the numerators over the common denominator.

$x^{2} + \frac{x}{2} = \frac{4 \cdot 16 - 9}{144}$

Simplify the numerator.

Multiply $4$ by $16$ .

$x^{2} + \frac{x}{2} = \frac{64 - 9}{144}$

Subtract $9$ from $64$ .

$x^{2} + \frac{x}{2} = \frac{55}{144}$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{1}{4})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + \frac{x}{2} + {(\frac{1}{4})}^{2} = \frac{55}{144} + {(\frac{1}{4})}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{1}{4}$ .

$x^{2} + \frac{x}{2} + \frac{1^{2}}{4^{2}} = \frac{55}{144} + {(\frac{1}{4})}^{2}$

One to any power is one.

$x^{2} + \frac{x}{2} + \frac{1}{4^{2}} = \frac{55}{144} + {(\frac{1}{4})}^{2}$

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

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{55}{144} + {(\frac{1}{4})}^{2}$

Simplify the right side.

Simplify $\frac{55}{144} + {(\frac{1}{4})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{1}{4}$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{55}{144} + \frac{1^{2}}{4^{2}}$

One to any power is one.

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{55}{144} + \frac{1}{4^{2}}$

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

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{55}{144} + \frac{1}{16}$

To write $\frac{1}{16}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{55}{144} + \frac{1}{16} \cdot \frac{9}{9}$

Write each expression with a common denominator of $144$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{16}$ by $\frac{9}{9}$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{55}{144} + \frac{9}{16 \cdot 9}$

Multiply $16$ by $9$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{55}{144} + \frac{9}{144}$

Simplify the expression.

Combine the numerators over the common denominator.

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{55 + 9}{144}$

Add $55$ and $9$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{64}{144}$

Cancel the common factor of $64$ and $144$ .

Factor $16$ out of $64$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{16 (4)}{144}$

Cancel the common factors.

Factor $16$ out of $144$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{16 \cdot 4}{16 \cdot 9}$

Cancel the common factor.

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{16 \cdot 4}{16 \cdot 9}$

Rewrite the expression.

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

Step 5

Factor the perfect trinomial square into ${(x + \frac{1}{4})}^{2}$ .

${(x + \frac{1}{4})}^{2} = \frac{4}{9}$

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.

$x + \frac{1}{4} = \pm \sqrt{\frac{4}{9}}$

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

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

$x + \frac{1}{4} = \pm \frac{\sqrt{4}}{\sqrt{9}}$

Simplify the numerator.

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

$x + \frac{1}{4} = \pm \frac{\sqrt{2^{2}}}{\sqrt{9}}$

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

$x + \frac{1}{4} = \pm \frac{2}{\sqrt{9}}$

Simplify the denominator.

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

$x + \frac{1}{4} = \pm \frac{2}{\sqrt{3^{2}}}$

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

$x + \frac{1}{4} = \pm \frac{2}{3}$

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.

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

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

Subtract $\frac{1}{4}$ from both sides of the equation.

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

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

$x = \frac{2}{3} \cdot \frac{4}{4} - \frac{1}{4}$

To write $- \frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{2}{3} \cdot \frac{4}{4} - \frac{1}{4} \cdot \frac{3}{3}$

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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

$x = \frac{2 \cdot 4}{3 \cdot 4} - \frac{1}{4} \cdot \frac{3}{3}$

Multiply $3$ by $4$ .

$x = \frac{2 \cdot 4}{12} - \frac{1}{4} \cdot \frac{3}{3}$

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

$x = \frac{2 \cdot 4}{12} - \frac{3}{4 \cdot 3}$

Multiply $4$ by $3$ .

$x = \frac{2 \cdot 4}{12} - \frac{3}{12}$

Combine the numerators over the common denominator.

$x = \frac{2 \cdot 4 - 3}{12}$

Simplify the numerator.

Multiply $2$ by $4$ .

$x = \frac{8 - 3}{12}$

Subtract $3$ from $8$ .

$x = \frac{5}{12}$

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

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

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

Subtract $\frac{1}{4}$ from both sides of the equation.

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

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

$x = - \frac{2}{3} \cdot \frac{4}{4} - \frac{1}{4}$

To write $- \frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - \frac{2}{3} \cdot \frac{4}{4} - \frac{1}{4} \cdot \frac{3}{3}$

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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

$x = - \frac{2 \cdot 4}{3 \cdot 4} - \frac{1}{4} \cdot \frac{3}{3}$

Multiply $3$ by $4$ .

$x = - \frac{2 \cdot 4}{12} - \frac{1}{4} \cdot \frac{3}{3}$

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

$x = - \frac{2 \cdot 4}{12} - \frac{3}{4 \cdot 3}$

Multiply $4$ by $3$ .

$x = - \frac{2 \cdot 4}{12} - \frac{3}{12}$

Combine the numerators over the common denominator.

$x = \frac{- 2 \cdot 4 - 3}{12}$

Simplify the numerator.

Multiply $- 2$ by $4$ .

$x = \frac{- 8 - 3}{12}$

Subtract $3$ from $- 8$ .

$x = \frac{- 11}{12}$

Move the negative in front of the fraction.

$x = - \frac{11}{12}$

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

$x = \frac{5}{12}, - \frac{11}{12}$

Solve by Completing the Square x^2+1/2x+1/16=4/9

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