Finite Math Examples

$f (x) = - 10 x - 8 + x (- 4 x + 9)$

Step 1

Set $- 10 x - 8 + x (- 4 x + 9)$ equal to $0$ .

$- 10 x - 8 + x (- 4 x + 9) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Simplify $- 10 x - 8 + x (- 4 x + 9)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$- 10 x - 8 + x (- 4 x) + x \cdot 9 = 0$

Step 2.1.1.2

Rewrite using the commutative property of multiplication.

$- 10 x - 8 - 4 x \cdot x + x \cdot 9 = 0$

Step 2.1.1.3

Move $9$ to the left of $x$ .

$- 10 x - 8 - 4 x \cdot x + 9 \cdot x = 0$

Step 2.1.1.4

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

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Step 2.1.1.4.1

Move $x$ .

$- 10 x - 8 - 4 (x \cdot x) + 9 \cdot x = 0$

Step 2.1.1.4.2

Multiply $x$ by $x$ .

$- 10 x - 8 - 4 x^{2} + 9 \cdot x = 0$

$- 10 x - 8 - 4 x^{2} + 9 x = 0$

Step 2.1.2

Add $- 10 x$ and $9 x$ .

$- x - 8 - 4 x^{2} = 0$

Step 2.2

Use the quadratic formula to find the solutions.

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

Step 2.3

Substitute the values $a = - 4$ , $b = - 1$ , and $c = - 8$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (- 4 \cdot - 8)}}{2 \cdot - 4}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot - 4 \cdot - 8}}{2 \cdot - 4}$

Step 2.4.1.2

Multiply $- 4 \cdot - 4 \cdot - 8$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $- 4$ .

$x = \frac{1 \pm \sqrt{1 + 16 \cdot - 8}}{2 \cdot - 4}$

Step 2.4.1.2.2

Multiply $16$ by $- 8$ .

$x = \frac{1 \pm \sqrt{1 - 128}}{2 \cdot - 4}$

Step 2.4.1.3

Subtract $128$ from $1$ .

$x = \frac{1 \pm \sqrt{- 127}}{2 \cdot - 4}$

Step 2.4.1.4

Rewrite $- 127$ as $- 1 (127)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 127}}{2 \cdot - 4}$

Step 2.4.1.5

Rewrite $\sqrt{- 1 (127)}$ as $\sqrt{- 1} \cdot \sqrt{127}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{127}}{2 \cdot - 4}$

Step 2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{1 \pm i \sqrt{127}}{2 \cdot - 4}$

Step 2.4.2

Multiply $2$ by $- 4$ .

$x = \frac{1 \pm i \sqrt{127}}{- 8}$

Step 2.4.3

Move the negative in front of the fraction.

$x = - \frac{1 \pm i \sqrt{127}}{8}$

Step 2.5

The final answer is the combination of both solutions.

$x = - \frac{1 + i \sqrt{127}}{8}, - \frac{1 - i \sqrt{127}}{8}$

$x = - \frac{1 \pm i \sqrt{127}}{8}$

Step 3