Precalculus Examples

$(11 x + 8) (3 x - 9) = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify $(11 x + 8) (3 x - 9)$ .

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Step 1.1.1

Expand $(11 x + 8) (3 x - 9)$ using the FOIL Method.

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Step 1.1.1.1

Apply the distributive property.

$11 x (3 x - 9) + 8 (3 x - 9) = 0$

Step 1.1.1.2

Apply the distributive property.

$11 x (3 x) + 11 x \cdot - 9 + 8 (3 x - 9) = 0$

Step 1.1.1.3

Apply the distributive property.

$11 x (3 x) + 11 x \cdot - 9 + 8 (3 x) + 8 \cdot - 9 = 0$

Step 1.1.2

Simplify and combine like terms.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

Rewrite using the commutative property of multiplication.

$11 \cdot 3 x \cdot x + 11 x \cdot - 9 + 8 (3 x) + 8 \cdot - 9 = 0$

Step 1.1.2.1.2

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

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Step 1.1.2.1.2.1

Move $x$ .

$11 \cdot 3 (x \cdot x) + 11 x \cdot - 9 + 8 (3 x) + 8 \cdot - 9 = 0$

Step 1.1.2.1.2.2

Multiply $x$ by $x$ .

$11 \cdot 3 x^{2} + 11 x \cdot - 9 + 8 (3 x) + 8 \cdot - 9 = 0$

Step 1.1.2.1.3

Multiply $11$ by $3$ .

$33 x^{2} + 11 x \cdot - 9 + 8 (3 x) + 8 \cdot - 9 = 0$

Step 1.1.2.1.4

Multiply $- 9$ by $11$ .

$33 x^{2} - 99 x + 8 (3 x) + 8 \cdot - 9 = 0$

Step 1.1.2.1.5

Multiply $3$ by $8$ .

$33 x^{2} - 99 x + 24 x + 8 \cdot - 9 = 0$

Step 1.1.2.1.6

Multiply $8$ by $- 9$ .

$33 x^{2} - 99 x + 24 x - 72 = 0$

Step 1.1.2.2

Add $- 99 x$ and $24 x$ .

$33 x^{2} - 75 x - 72 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 33$ , $b = - 75$ , and $c = - 72$ into the quadratic formula and solve for $x$ .

$\frac{75 \pm \sqrt{{(- 75)}^{2} - 4 \cdot (33 \cdot - 72)}}{2 \cdot 33}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 75$ to the power of $2$ .

$x = \frac{75 \pm \sqrt{5625 - 4 \cdot 33 \cdot - 72}}{2 \cdot 33}$

Step 4.1.2

Multiply $- 4 \cdot 33 \cdot - 72$ .

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Step 4.1.2.1

Multiply $- 4$ by $33$ .

$x = \frac{75 \pm \sqrt{5625 - 132 \cdot - 72}}{2 \cdot 33}$

Step 4.1.2.2

Multiply $- 132$ by $- 72$ .

$x = \frac{75 \pm \sqrt{5625 + 9504}}{2 \cdot 33}$

Step 4.1.3

Add $5625$ and $9504$ .

$x = \frac{75 \pm \sqrt{15129}}{2 \cdot 33}$

Step 4.1.4

Rewrite $15129$ as $123^{2}$ .

$x = \frac{75 \pm \sqrt{123^{2}}}{2 \cdot 33}$

Step 4.1.5

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

$x = \frac{75 \pm 123}{2 \cdot 33}$

Step 4.2

Multiply $2$ by $33$ .

$x = \frac{75 \pm 123}{66}$

Step 4.3

Simplify $\frac{75 \pm 123}{66}$ .

$x = \frac{25 \pm 41}{22}$

Step 5

The final answer is the combination of both solutions.

$x = 3, - \frac{8}{11}$