Precalculus Examples

$3 x^{2} - 12 x + 6 = 0$

Step 1

Subtract $6$ from both sides of the equation.

$3 x^{2} - 12 x = - 6$

Step 2

Divide each term in $3 x^{2} - 12 x = - 6$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 x^{2} - 12 x = - 6$ by $3$ .

$\frac{3 x^{2}}{3} + \frac{- 12 x}{3} = \frac{- 6}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $3$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{- 12 x}{3} = \frac{- 6}{3}$

Step 2.2.1.1.2

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

$x^{2} + \frac{- 12 x}{3} = \frac{- 6}{3}$

Step 2.2.1.2

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

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Step 2.2.1.2.1

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

$x^{2} + \frac{3 (- 4 x)}{3} = \frac{- 6}{3}$

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor $3$ out of $3$ .

$x^{2} + \frac{3 (- 4 x)}{3 (1)} = \frac{- 6}{3}$

Step 2.2.1.2.2.2

Cancel the common factor.

$x^{2} + \frac{3 (- 4 x)}{3 \cdot 1} = \frac{- 6}{3}$

Step 2.2.1.2.2.3

Rewrite the expression.

$x^{2} + \frac{- 4 x}{1} = \frac{- 6}{3}$

Step 2.2.1.2.2.4

Divide $- 4 x$ by $1$ .

$x^{2} - 4 x = \frac{- 6}{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 6$ by $3$ .

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

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.

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

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

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

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

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 2 + {(- 2)}^{2}$ .

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Step 5.2.1.1

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

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

Step 5.2.1.2

Add $- 2$ and $4$ .

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

Step 6

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

${(x - 2)}^{2} = 2$

Step 7

Solve the equation for $x$ .

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Step 7.1

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

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

Step 7.2

Add $2$ to both sides of the equation.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 3.41421356 \dots, 0.58578643 \dots$