Solve by Completing the Square 3x^2-6x-1=0

$3 x^{2} - 6 x - 1 = 0$

Step 1

Add $1$ to both sides of the equation.

$3 x^{2} - 6 x = 1$

Step 2

Divide each term in $3 x^{2} - 6 x = 1$ by $3$ and simplify.

Divide each term in $3 x^{2} - 6 x = 1$ by $3$ .

$\frac{3 x^{2}}{3} + \frac{- 6 x}{3} = \frac{1}{3}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{- 6 x}{3} = \frac{1}{3}$

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

$x^{2} + \frac{- 6 x}{3} = \frac{1}{3}$

Cancel the common factor of $- 6$ and $3$ .

Factor $3$ out of $- 6 x$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 2 x$ by $1$ .

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

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 $b$ .

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

Step 4

Add the term to each side of the equation.

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

Step 5

Simplify the equation.

Simplify the left side.

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

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

Simplify the right side.

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $1$ and $3$ .

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

Step 6

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

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

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.

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

Simplify $\pm \sqrt{\frac{4}{3}}$ .

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

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

Simplify the numerator.

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

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

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Add $1$ to both sides of the equation.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2.15470053 \dots, - 0.15470053 \dots$

Solve by Completing the Square 3x^2-6x-1=0

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