Precalculus Examples

$\frac{3}{4} x^{2} + 8 x + 20 = 0$

Step 1

Combine $\frac{3}{4}$ and $x^{2}$ .

$\frac{3 x^{2}}{4} + 8 x + 20 = 0$

Step 2

To write $8 x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3 x^{2}}{4} + 8 x \cdot \frac{4}{4} + 20 = 0$

Step 3

Combine $8 x$ and $\frac{4}{4}$ .

$\frac{3 x^{2}}{4} + \frac{8 x \cdot 4}{4} + 20 = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{3 x^{2} + 8 x \cdot 4}{4} + 20 = 0$

Step 5

Simplify the numerator.

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Step 5.1

Factor $x$ out of $3 x^{2} + 8 x \cdot 4$ .

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Step 5.1.1

Factor $x$ out of $3 x^{2}$ .

$\frac{x (3 x) + 8 x \cdot 4}{4} + 20 = 0$

Step 5.1.2

Factor $x$ out of $8 x \cdot 4$ .

$\frac{x (3 x) + x (8 \cdot 4)}{4} + 20 = 0$

Step 5.1.3

Factor $x$ out of $x (3 x) + x (8 \cdot 4)$ .

$\frac{x (3 x + 8 \cdot 4)}{4} + 20 = 0$

Step 5.2

Multiply $8$ by $4$ .

$\frac{x (3 x + 32)}{4} + 20 = 0$

Step 6

To write $20$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x (3 x + 32)}{4} + 20 \cdot \frac{4}{4} = 0$

Step 7

Combine $20$ and $\frac{4}{4}$ .

$\frac{x (3 x + 32)}{4} + \frac{20 \cdot 4}{4} = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{x (3 x + 32) + 20 \cdot 4}{4} = 0$

Step 9

Simplify the numerator.

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Step 9.1

Apply the distributive property.

$\frac{x (3 x) + x \cdot 32 + 20 \cdot 4}{4} = 0$

Step 9.2

Rewrite using the commutative property of multiplication.

$\frac{3 x \cdot x + x \cdot 32 + 20 \cdot 4}{4} = 0$

Step 9.3

Move $32$ to the left of $x$ .

$\frac{3 x \cdot x + 32 \cdot x + 20 \cdot 4}{4} = 0$

Step 9.4

Multiply $x$ by $x$ by adding the exponents.

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Step 9.4.1

Move $x$ .

$\frac{3 (x \cdot x) + 32 \cdot x + 20 \cdot 4}{4} = 0$

Step 9.4.2

Multiply $x$ by $x$ .

$\frac{3 x^{2} + 32 \cdot x + 20 \cdot 4}{4} = 0$

$\frac{3 x^{2} + 32 x + 20 \cdot 4}{4} = 0$

Step 9.5

Multiply $20$ by $4$ .

$\frac{3 x^{2} + 32 x + 80}{4} = 0$

Step 9.6

Factor by grouping.

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Step 9.6.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 80 = 240$ and whose sum is $b = 32$ .

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Step 9.6.1.1

Factor $32$ out of $32 x$ .

$\frac{3 x^{2} + 32 (x) + 80}{4} = 0$

Step 9.6.1.2

Rewrite $32$ as $12$ plus $20$

$\frac{3 x^{2} + (12 + 20) x + 80}{4} = 0$

Step 9.6.1.3

Apply the distributive property.

$\frac{3 x^{2} + 12 x + 20 x + 80}{4} = 0$

Step 9.6.2

Factor out the greatest common factor from each group.

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Step 9.6.2.1

Group the first two terms and the last two terms.

$\frac{(3 x^{2} + 12 x) + 20 x + 80}{4} = 0$

Step 9.6.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{3 x (x + 4) + 20 (x + 4)}{4} = 0$

Step 9.6.3

Factor the polynomial by factoring out the greatest common factor, $x + 4$ .

$\frac{(x + 4) (3 x + 20)}{4} = 0$

Step 10

Set the numerator equal to zero.

$(x + 4) (3 x + 20) = 0$

Step 11

Solve the equation for $x$ .

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Step 11.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 4 = 0$

$3 x + 20 = 0$

Step 11.2

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 11.2.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 11.2.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 11.3

Set $3 x + 20$ equal to $0$ and solve for $x$ .

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Step 11.3.1

Set $3 x + 20$ equal to $0$ .

$3 x + 20 = 0$

Step 11.3.2

Solve $3 x + 20 = 0$ for $x$ .

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Step 11.3.2.1

Subtract $20$ from both sides of the equation.

$3 x = - 20$

Step 11.3.2.2

Divide each term in $3 x = - 20$ by $3$ and simplify.

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Step 11.3.2.2.1

Divide each term in $3 x = - 20$ by $3$ .

$\frac{3 x}{3} = \frac{- 20}{3}$

Step 11.3.2.2.2

Simplify the left side.

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Step 11.3.2.2.2.1

Cancel the common factor of $3$ .

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Step 11.3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 20}{3}$

Step 11.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 20}{3}$

Step 11.3.2.2.3

Simplify the right side.

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Step 11.3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{20}{3}$

Step 11.4

The final solution is all the values that make $(x + 4) (3 x + 20) = 0$ true.

$x = - 4, - \frac{20}{3}$