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

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

Step 1

Add $\frac{1}{4}$ to both sides of the equation.

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

Step 2

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{1}{4} \cdot \frac{1}{2}$ .

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

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

Multiply $4$ by $2$ .

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

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 $b$ .

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

Step 4

Add the term to each side of the equation.

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

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

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

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

One to any power is one.

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

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

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{1}{8} + {(\frac{1}{4})}^{2}$

Simplify the right side.

Simplify $\frac{1}{8} + {(\frac{1}{4})}^{2}$ .

Simplify each term.

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

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{1}{8} + \frac{1^{2}}{4^{2}}$

One to any power is one.

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{1}{8} + \frac{1}{4^{2}}$

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

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{1}{8} + \frac{1}{16}$

To write $\frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{1}{8} \cdot \frac{2}{2} + \frac{1}{16}$

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{2}{8 \cdot 2} + \frac{1}{16}$

Multiply $8$ by $2$ .

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

Combine the numerators over the common denominator.

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

Add $2$ and $1$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{3}{16}$

Step 6

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

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

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.

$x + \frac{1}{4} = \pm \sqrt{\frac{3}{16}}$

Simplify $\pm \sqrt{\frac{3}{16}}$ .

Rewrite $\sqrt{\frac{3}{16}}$ as $\frac{\sqrt{3}}{\sqrt{16}}$ .

$x + \frac{1}{4} = \pm \frac{\sqrt{3}}{\sqrt{16}}$

Simplify the denominator.

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

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

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

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

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

$x = \pm \frac{\sqrt{3}}{4} - \frac{1}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{3}}{4} - \frac{1}{4}$

Decimal Form:

$x = 0.18301270 \dots, - 0.68301270 \dots$

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

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