Solve by Completing the Square 3x^2-12x+1=0

$3 x^{2} - 12 x + 1 = 0$

Step 1

Subtract $1$ from both sides of the equation.

$3 x^{2} - 12 x = - 1$

Step 2

Divide each term in $3 x^{2} - 12 x = - 1$ by $3$ and simplify.

Divide each term in $3 x^{2} - 12 x = - 1$ by $3$ .

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

Simplify the left side.

Simplify each term.

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

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

Cancel the common factor of $- 12$ and $3$ .

Factor $3$ out of $- 12 x$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 4 x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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 $b$ .

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

Step 4

Add the term to each side of the equation.

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

Step 5

Simplify the equation.

Simplify the left side.

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

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

Simplify the right side.

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

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

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

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

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

Combine $4$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $4$ by $3$ .

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

Add $- 1$ and $12$ .

$x^{2} - 4 x + 4 = \frac{11}{3}$

Step 6

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

${(x - 2)}^{2} = \frac{11}{3}$

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.

$x - 2 = \pm \sqrt{\frac{11}{3}}$

Simplify $\pm \sqrt{\frac{11}{3}}$ .

Rewrite $\sqrt{\frac{11}{3}}$ as $\frac{\sqrt{11}}{\sqrt{3}}$ .

$x - 2 = \pm \frac{\sqrt{11}}{\sqrt{3}}$

Multiply $\frac{\sqrt{11}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x - 2 = \pm \frac{\sqrt{11}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{11}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x - 2 = \pm \frac{\sqrt{11} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$x - 2 = \pm \frac{\sqrt{11} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

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

$x - 2 = \pm \frac{\sqrt{11} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

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

$x - 2 = \pm \frac{\sqrt{11} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$x - 2 = \pm \frac{\sqrt{11} \sqrt{3}}{{\sqrt{3}}^{2}}$

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

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

$x - 2 = \pm \frac{\sqrt{11} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$x - 2 = \pm \frac{\sqrt{11} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

$x - 2 = \pm \frac{\sqrt{11} \sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x - 2 = \pm \frac{\sqrt{11} \sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$x - 2 = \pm \frac{\sqrt{11} \sqrt{3}}{3^{1}}$

Evaluate the exponent.

$x - 2 = \pm \frac{\sqrt{11} \sqrt{3}}{3}$

Simplify the numerator.

Combine using the product rule for radicals.

$x - 2 = \pm \frac{\sqrt{11 \cdot 3}}{3}$

Multiply $11$ by $3$ .

$x - 2 = \pm \frac{\sqrt{33}}{3}$

Add $2$ to both sides of the equation.

$x = \pm \frac{\sqrt{33}}{3} + 2$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{33}}{3} + 2$

Decimal Form:

$x = 3.91485421 \dots, 0.08514578 \dots$

Solve by Completing the Square 3x^2-12x+1=0

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