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

$4 x^{2} - 4 x - 1 = 0$

Step 1

Add $1$ to both sides of the equation.

$4 x^{2} - 4 x = 1$

Step 2

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

Divide each term in $4 x^{2} - 4 x = 1$ by $4$ .

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

Simplify the left side.

Simplify each term.

Cancel the common factor of $4$ .

Cancel the common factor.

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

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

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

Cancel the common factor of $- 4$ and $4$ .

Factor $4$ out of $- 4 x$ .

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

Cancel the common factors.

Factor $4$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- x$ by $1$ .

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

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 $b$ .

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

Step 4

Add the term to each side of the equation.

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

Step 5

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 $- \frac{1}{2}$ .

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

Apply the product rule to $\frac{1}{2}$ .

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

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

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

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

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

One to any power is one.

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

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

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

Simplify the right side.

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

Simplify each term.

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

Apply the product rule to $- \frac{1}{2}$ .

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

Apply the product rule to $\frac{1}{2}$ .

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

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

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

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

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

One to any power is one.

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

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

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

Simplify terms.

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

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

${(x - \frac{1}{2})}^{2} = \frac{1}{2}$

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.

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

Simplify $\pm \sqrt{\frac{1}{2}}$ .

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Any root of $1$ is $1$ .

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

$x - \frac{1}{2} = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Add $\frac{1}{2}$ to both sides of the equation.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.20710678 \dots, - 0.20710678 \dots$

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

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