Finite Math Examples

$3 \cdot (x + 1) \cdot (x - 6)$

Step 1

Set $3 \cdot (x + 1) \cdot (x - 6)$ equal to $0$ .

$3 \cdot (x + 1) \cdot (x - 6) = 0$

Step 2

Simplify the left side.

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Step 2.1

Simplify $3 \cdot (x + 1) \cdot (x - 6)$ .

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Step 2.1.1

Simplify by multiplying through.

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Step 2.1.1.1

Apply the distributive property.

$(3 x + 3 \cdot 1) \cdot (x - 6) = 0$

Step 2.1.1.2

Multiply $3$ by $1$ .

$(3 x + 3) \cdot (x - 6) = 0$

Step 2.1.2

Expand $(3 x + 3) (x - 6)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$3 x (x - 6) + 3 (x - 6) = 0$

Step 2.1.2.2

Apply the distributive property.

$3 x \cdot x + 3 x \cdot - 6 + 3 (x - 6) = 0$

Step 2.1.2.3

Apply the distributive property.

$3 x \cdot x + 3 x \cdot - 6 + 3 x + 3 \cdot - 6 = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

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

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Step 2.1.3.1.1.1

Move $x$ .

$3 (x \cdot x) + 3 x \cdot - 6 + 3 x + 3 \cdot - 6 = 0$

Step 2.1.3.1.1.2

Multiply $x$ by $x$ .

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

Step 2.1.3.1.2

Multiply $- 6$ by $3$ .

$3 x^{2} - 18 x + 3 x + 3 \cdot - 6 = 0$

Step 2.1.3.1.3

Multiply $3$ by $- 6$ .

$3 x^{2} - 18 x + 3 x - 18 = 0$

Step 2.1.3.2

Add $- 18 x$ and $3 x$ .

$3 x^{2} - 15 x - 18 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 3$ , $b = - 15$ , and $c = - 18$ into the quadratic formula and solve for $x$ .

$\frac{15 \pm \sqrt{{(- 15)}^{2} - 4 \cdot (3 \cdot - 18)}}{2 \cdot 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 3 \cdot - 18}}{2 \cdot 3}$

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot - 18$ .

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{15 \pm \sqrt{225 - 12 \cdot - 18}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $- 18$ .

$x = \frac{15 \pm \sqrt{225 + 216}}{2 \cdot 3}$

Step 5.1.3

Add $225$ and $216$ .

$x = \frac{15 \pm \sqrt{441}}{2 \cdot 3}$

Step 5.1.4

Rewrite $441$ as $21^{2}$ .

$x = \frac{15 \pm \sqrt{21^{2}}}{2 \cdot 3}$

Step 5.1.5

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

$x = \frac{15 \pm 21}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$x = \frac{15 \pm 21}{6}$

Step 5.3

Simplify $\frac{15 \pm 21}{6}$ .

$x = \frac{5 \pm 7}{2}$

Step 6

The final answer is the combination of both solutions.

$x = 6, - 1$