Solve by Completing the Square 4x^2+x-1/16=0

$4 x^{2} + x - \frac{1}{16} = 0$

Step 1

Add $\frac{1}{16}$ to both sides of the equation.

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

Step 2

Divide each term in $4 x^{2} + x = \frac{1}{16}$ by $4$ and simplify.

Divide each term in $4 x^{2} + x = \frac{1}{16}$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{1}{16} \cdot \frac{1}{4}$ .

Multiply $\frac{1}{16}$ by $\frac{1}{4}$ .

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

Multiply $16$ by $4$ .

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

Step 3

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 4

Add the term to each side of the equation.

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

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

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

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

One to any power is one.

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

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

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

Simplify the right side.

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

Simplify each term.

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

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

One to any power is one.

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

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

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

Simplify terms.

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Cancel the common factor of $2$ and $64$ .

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $64$ .

$x^{2} + \frac{x}{4} + \frac{1}{64} = \frac{2 \cdot 1}{2 \cdot 32}$

Cancel the common factor.

$x^{2} + \frac{x}{4} + \frac{1}{64} = \frac{2 \cdot 1}{2 \cdot 32}$

Rewrite the expression.

$x^{2} + \frac{x}{4} + \frac{1}{64} = \frac{1}{32}$

Step 6

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

${(x + \frac{1}{8})}^{2} = \frac{1}{32}$

Step 7

Solve the equation for $x$ .

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

$x + \frac{1}{8} = \pm \sqrt{\frac{1}{32}}$

Simplify $\pm \sqrt{\frac{1}{32}}$ .

Rewrite $\sqrt{\frac{1}{32}}$ as $\frac{\sqrt{1}}{\sqrt{32}}$ .

$x + \frac{1}{8} = \pm \frac{\sqrt{1}}{\sqrt{32}}$

Any root of $1$ is $1$ .

$x + \frac{1}{8} = \pm \frac{1}{\sqrt{32}}$

Simplify the denominator.

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$x + \frac{1}{8} = \pm \frac{1}{\sqrt{16 (2)}}$

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

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

Pull terms out from under the radical.

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

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

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

Combine and simplify the denominator.

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

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

Move $\sqrt{2}$ .

$x + \frac{1}{8} = \pm \frac{\sqrt{2}}{4 (\sqrt{2} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$x + \frac{1}{8} = \pm \frac{\sqrt{2}}{4 ({\sqrt{2}}^{1} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$x + \frac{1}{8} = \pm \frac{\sqrt{2}}{4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x + \frac{1}{8} = \pm \frac{\sqrt{2}}{4 {\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$x + \frac{1}{8} = \pm \frac{\sqrt{2}}{4 {\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x + \frac{1}{8} = \pm \frac{\sqrt{2}}{4 {(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $4$ by $2$ .

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

Subtract $\frac{1}{8}$ from both sides of the equation.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.05177669 \dots, - 0.30177669 \dots$

Solve by Completing the Square 4x^2+x-1/16=0

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