Precalculus Examples

$- 6 x^{2} + 12 x + 378 = 0$

Step 1

Subtract $378$ from both sides of the equation.

$- 6 x^{2} + 12 x = - 378$

Step 2

Divide each term in $- 6 x^{2} + 12 x = - 378$ by $- 6$ and simplify.

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Step 2.1

Divide each term in $- 6 x^{2} + 12 x = - 378$ by $- 6$ .

$\frac{- 6 x^{2}}{- 6} + \frac{12 x}{- 6} = \frac{- 378}{- 6}$

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 $- 6$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{- 6 x^{2}}{- 6} + \frac{12 x}{- 6} = \frac{- 378}{- 6}$

Step 2.2.1.1.2

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

$x^{2} + \frac{12 x}{- 6} = \frac{- 378}{- 6}$

Step 2.2.1.2

Cancel the common factor of $12$ and $- 6$ .

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Step 2.2.1.2.1

Factor $6$ out of $12 x$ .

$x^{2} + \frac{6 (2 x)}{- 6} = \frac{- 378}{- 6}$

Step 2.2.1.2.2

Move the negative one from the denominator of $\frac{2 x}{- 1}$ .

$x^{2} - 1 \cdot (2 x) = \frac{- 378}{- 6}$

Step 2.2.1.3

Rewrite $- 1 \cdot (2 x)$ as $- (2 x)$ .

$x^{2} - (2 x) = \frac{- 378}{- 6}$

Step 2.2.1.4

Multiply $2$ by $- 1$ .

$x^{2} - 2 x = \frac{- 378}{- 6}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 378$ by $- 6$ .

$x^{2} - 2 x = 63$

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.

$x^{2} - 2 x + {(- 1)}^{2} = 63 + {(- 1)}^{2}$

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

$x^{2} - 2 x + 1 = 63 + {(- 1)}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $63 + {(- 1)}^{2}$ .

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Step 5.2.1.1

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

$x^{2} - 2 x + 1 = 63 + 1$

Step 5.2.1.2

Add $63$ and $1$ .

$x^{2} - 2 x + 1 = 64$

Step 6

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

${(x - 1)}^{2} = 64$

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.

$x - 1 = \pm \sqrt{64}$

Step 7.2

Simplify $\pm \sqrt{64}$ .

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Step 7.2.1

Rewrite $64$ as $8^{2}$ .

$x - 1 = \pm \sqrt{8^{2}}$

Step 7.2.2

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

$x - 1 = \pm 8$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x - 1 = 8$

Step 7.3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 7.3.2.1

Add $1$ to both sides of the equation.

$x = 8 + 1$

Step 7.3.2.2

Add $8$ and $1$ .

$x = 9$

Step 7.3.3

Next, use the negative value of the $\pm$ to find the second solution.

$x - 1 = - 8$

Step 7.3.4

Move all terms not containing $x$ to the right side of the equation.

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Step 7.3.4.1

Add $1$ to both sides of the equation.

$x = - 8 + 1$

Step 7.3.4.2

Add $- 8$ and $1$ .

$x = - 7$

Step 7.3.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = 9, - 7$