Solve by Completing the Square 3x^2-3x-1=0

$3 x^{2} - 3 x - 1 = 0$

Add $1$ to both sides of the equation.

$3 x^{2} - 3 x = 1$

Divide each term in $3 x^{2} - 3 x = 1$ by $3$ and simplify.

Divide each term in $3 x^{2} - 3 x = 1$ by $3$ .

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

Simplify the left side.

Simplify each term.

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

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

Cancel the common factor of $- 3$ and $3$ .

Factor $3$ out of $- 3 x$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

$x^{2} + \frac{3 (- x)}{3 \cdot 1} = \frac{1}{3}$

Rewrite the expression.

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

Divide $- x$ by $1$ .

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

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

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

Add the term to each side of the equation.

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

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

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

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

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

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

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

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

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

One to any power is one.

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

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

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

Simplify the right side.

Simplify $\frac{1}{3} + {(- \frac{1}{2})}^{2}$ .

Simplify each term.

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

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

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

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

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

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

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

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

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

One to any power is one.

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

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

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

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

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

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

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

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

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

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

Multiply $3$ by $4$ .

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

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

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

Multiply $4$ by $3$ .

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

Combine the numerators over the common denominator.

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

Add $4$ and $3$ .

$x^{2} - x + \frac{1}{4} = \frac{7}{12}$

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

${(x - \frac{1}{2})}^{2} = \frac{7}{12}$

Solve the equation for $x$ .

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

$x - \frac{1}{2} = \pm \sqrt{\frac{7}{12}}$

Simplify $\pm \sqrt{\frac{7}{12}}$ .

Rewrite $\sqrt{\frac{7}{12}}$ as $\frac{\sqrt{7}}{\sqrt{12}}$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{7}}{\sqrt{12}}$

Simplify the denominator.

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{7}}{\sqrt{4 (3)}}$

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

$x - \frac{1}{2} = \pm \frac{\sqrt{7}}{\sqrt{2^{2} \cdot 3}}$

Pull terms out from under the radical.

$x - \frac{1}{2} = \pm \frac{\sqrt{7}}{2 \sqrt{3}}$

Multiply $\frac{\sqrt{7}}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{7}}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{7}}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{7} \sqrt{3}}{2 \sqrt{3} \sqrt{3}}$

Move $\sqrt{3}$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{7} \sqrt{3}}{2 (\sqrt{3} \sqrt{3})}$

Raise $\sqrt{3}$ to the power of $1$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{7} \sqrt{3}}{2 ({\sqrt{3}}^{1} \sqrt{3})}$

Raise $\sqrt{3}$ to the power of $1$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{7} \sqrt{3}}{2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

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

$x - \frac{1}{2} = \pm \frac{\sqrt{7} \sqrt{3}}{2 {\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{7} \sqrt{3}}{2 {\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{7} \sqrt{3}}{2 {(3^{\frac{1}{2}})}^{2}}$

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

$x - \frac{1}{2} = \pm \frac{\sqrt{7} \sqrt{3}}{2 \cdot 3^{\frac{1}{2} \cdot 2}}$

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

$x - \frac{1}{2} = \pm \frac{\sqrt{7} \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x - \frac{1}{2} = \pm \frac{\sqrt{7} \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Rewrite the expression.

$x - \frac{1}{2} = \pm \frac{\sqrt{7} \sqrt{3}}{2 \cdot 3^{1}}$

Evaluate the exponent.

$x - \frac{1}{2} = \pm \frac{\sqrt{7} \sqrt{3}}{2 \cdot 3}$

Simplify the numerator.

Combine using the product rule for radicals.

$x - \frac{1}{2} = \pm \frac{\sqrt{7 \cdot 3}}{2 \cdot 3}$

Multiply $7$ by $3$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{21}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{21}}{6}$

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

$x = \pm \frac{\sqrt{21}}{6} + \frac{1}{2}$

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{21}}{6} + \frac{1}{2}$

Decimal Form:

$x = 1.26376261 \dots, - 0.26376261 \dots$

Solve by Completing the Square 3x^2-3x-1=0

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