Finite Math Examples

$f (x) = - 2 x^{2} + 8 x + 10$

Step 1

Set $- 2 x^{2} + 8 x + 10$ equal to $0$ .

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

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $- 2$ out of $- 2 x^{2} + 8 x + 10$ .

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Step 2.1.1.1

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

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

Step 2.1.1.2

Factor $- 2$ out of $8 x$ .

$- 2 x^{2} - 2 (- 4 x) + 10 = 0$

Step 2.1.1.3

Factor $- 2$ out of $10$ .

$- 2 x^{2} - 2 (- 4 x) - 2 \cdot - 5 = 0$

Step 2.1.1.4

Factor $- 2$ out of $- 2 (x^{2}) - 2 (- 4 x)$ .

$- 2 (x^{2} - 4 x) - 2 \cdot - 5 = 0$

Step 2.1.1.5

Factor $- 2$ out of $- 2 (x^{2} - 4 x) - 2 (- 5)$ .

$- 2 (x^{2} - 4 x - 5) = 0$

Step 2.1.2

Factor.

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Step 2.1.2.1

Factor $x^{2} - 4 x - 5$ using the AC method.

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Step 2.1.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 2.1.2.1.2

Write the factored form using these integers.

$- 2 ((x - 5) (x + 1)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$- 2 (x - 5) (x + 1) = 0$

Step 2.2

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

$x - 5 = 0$

$x + 1 = 0$

Step 2.3

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.5

The final solution is all the values that make $- 2 (x - 5) (x + 1) = 0$ true.

$x = 5, - 1$

Step 3