Precalculus Examples

$3 g^{2} - 12 g = - 4$

Step 1

Divide each term in $3 g^{2} - 12 g = - 4$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 g^{2} - 12 g = - 4$ by $3$ .

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Cancel the common factor of $3$ .

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Step 1.2.1.1.1

Cancel the common factor.

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

Step 1.2.1.1.2

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

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

Step 1.2.1.2

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

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Step 1.2.1.2.1

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

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

Step 1.2.1.2.2

Cancel the common factors.

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Step 1.2.1.2.2.1

Factor $3$ out of $3$ .

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

Step 1.2.1.2.2.2

Cancel the common factor.

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

Step 1.2.1.2.2.3

Rewrite the expression.

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

Step 1.2.1.2.2.4

Divide $- 4 g$ by $1$ .

$g^{2} - 4 g = \frac{- 4}{3}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$g^{2} - 4 g = - \frac{4}{3}$

Step 2

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 3

Add the term to each side of the equation.

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

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

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

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

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{4}{3} + {(- 2)}^{2}$ .

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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

Step 4.2.1.4

Combine the numerators over the common denominator.

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

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $4$ by $3$ .

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

Step 4.2.1.5.2

Add $- 4$ and $12$ .

$g^{2} - 4 g + 4 = \frac{8}{3}$

Step 5

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

${(g - 2)}^{2} = \frac{8}{3}$

Step 6

Solve the equation for $g$ .

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Step 6.1

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

$g - 2 = \pm \sqrt{\frac{8}{3}}$

Step 6.2

Simplify $\pm \sqrt{\frac{8}{3}}$ .

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Step 6.2.1

Rewrite $\sqrt{\frac{8}{3}}$ as $\frac{\sqrt{8}}{\sqrt{3}}$ .

$g - 2 = \pm \frac{\sqrt{8}}{\sqrt{3}}$

Step 6.2.2

Simplify the numerator.

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Step 6.2.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.2.2.1.1

Factor $4$ out of $8$ .

$g - 2 = \pm \frac{\sqrt{4 (2)}}{\sqrt{3}}$

Step 6.2.2.1.2

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

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

Step 6.2.2.2

Pull terms out from under the radical.

$g - 2 = \pm \frac{2 \sqrt{2}}{\sqrt{3}}$

Step 6.2.3

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

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

Step 6.2.4

Combine and simplify the denominator.

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Step 6.2.4.1

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

$g - 2 = \pm \frac{2 \sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 6.2.4.2

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

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

Step 6.2.4.3

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

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

Step 6.2.4.4

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

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

Step 6.2.4.5

Add $1$ and $1$ .

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

Step 6.2.4.6

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

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Step 6.2.4.6.1

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

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

Step 6.2.4.6.2

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

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

Step 6.2.4.6.3

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

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

Step 6.2.4.6.4

Cancel the common factor of $2$ .

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Step 6.2.4.6.4.1

Cancel the common factor.

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

Step 6.2.4.6.4.2

Rewrite the expression.

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

Step 6.2.4.6.5

Evaluate the exponent.

$g - 2 = \pm \frac{2 \sqrt{2} \sqrt{3}}{3}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Combine using the product rule for radicals.

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

Step 6.2.5.2

Multiply $3$ by $2$ .

$g - 2 = \pm \frac{2 \sqrt{6}}{3}$

Step 6.3

Add $2$ to both sides of the equation.

$g = \pm \frac{2 \sqrt{6}}{3} + 2$

Step 7

The result can be shown in multiple forms.

Exact Form:

$g = \pm \frac{2 \sqrt{6}}{3} + 2$

Decimal Form:

$g = 3.63299316 \dots, 0.36700683 \dots$