Finite Math Examples

f (x) = 3 {(x - 8)}^{2} - 27

$f (x) = 3 {(x - 8)}^{2} - 27$

Step 1

Set

3 {(x - 8)}^{2} - 27

$3 {(x - 8)}^{2} - 27$ equal to

0

$0$ .

3 {(x - 8)}^{2} - 27 = 0

$3 {(x - 8)}^{2} - 27 = 0$

Step 2

Solve for

x

$x$ .

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Step 2.1

Simplify

3 {(x - 8)}^{2} - 27

$3 {(x - 8)}^{2} - 27$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Rewrite

{(x - 8)}^{2}

${(x - 8)}^{2}$ as

(x - 8) (x - 8)

$(x - 8) (x - 8)$ .

3 ((x - 8) (x - 8)) - 27 = 0

$3 ((x - 8) (x - 8)) - 27 = 0$

Step 2.1.1.2

Expand

(x - 8) (x - 8)

$(x - 8) (x - 8)$ using the FOIL Method.

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Step 2.1.1.2.1

Apply the distributive property.

3 (x (x - 8) - 8 (x - 8)) - 27 = 0

$3 (x (x - 8) - 8 (x - 8)) - 27 = 0$

Step 2.1.1.2.2

Apply the distributive property.

3 (x \cdot x + x \cdot - 8 - 8 (x - 8)) - 27 = 0

$3 (x \cdot x + x \cdot - 8 - 8 (x - 8)) - 27 = 0$

Step 2.1.1.2.3

Apply the distributive property.

3 (x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8) - 27 = 0

$3 (x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8) - 27 = 0$

3 (x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8) - 27 = 0

$3 (x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8) - 27 = 0$

Step 2.1.1.3

Simplify and combine like terms.

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Step 2.1.1.3.1

Simplify each term.

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Step 2.1.1.3.1.1

Multiply

x

$x$ by

x

$x$ .

3 (x^{2} + x \cdot - 8 - 8 x - 8 \cdot - 8) - 27 = 0

$3 (x^{2} + x \cdot - 8 - 8 x - 8 \cdot - 8) - 27 = 0$

Step 2.1.1.3.1.2

Move

- 8

$- 8$ to the left of

x

$x$ .

3 (x^{2} - 8 \cdot x - 8 x - 8 \cdot - 8) - 27 = 0

$3 (x^{2} - 8 \cdot x - 8 x - 8 \cdot - 8) - 27 = 0$

Step 2.1.1.3.1.3

Multiply

- 8

$- 8$ by

- 8

$- 8$ .

3 (x^{2} - 8 x - 8 x + 64) - 27 = 0

$3 (x^{2} - 8 x - 8 x + 64) - 27 = 0$

3 (x^{2} - 8 x - 8 x + 64) - 27 = 0

$3 (x^{2} - 8 x - 8 x + 64) - 27 = 0$

Step 2.1.1.3.2

Subtract

8 x

$8 x$ from

- 8 x

$- 8 x$ .

3 (x^{2} - 16 x + 64) - 27 = 0

$3 (x^{2} - 16 x + 64) - 27 = 0$

3 (x^{2} - 16 x + 64) - 27 = 0

$3 (x^{2} - 16 x + 64) - 27 = 0$

Step 2.1.1.4

Apply the distributive property.

3 x^{2} + 3 (- 16 x) + 3 \cdot 64 - 27 = 0

$3 x^{2} + 3 (- 16 x) + 3 \cdot 64 - 27 = 0$

Step 2.1.1.5

Simplify.

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Step 2.1.1.5.1

Multiply

- 16

$- 16$ by

3

$3$ .

3 x^{2} - 48 x + 3 \cdot 64 - 27 = 0

$3 x^{2} - 48 x + 3 \cdot 64 - 27 = 0$

Step 2.1.1.5.2

Multiply

3

$3$ by

64

$64$ .

3 x^{2} - 48 x + 192 - 27 = 0

$3 x^{2} - 48 x + 192 - 27 = 0$

3 x^{2} - 48 x + 192 - 27 = 0

$3 x^{2} - 48 x + 192 - 27 = 0$

3 x^{2} - 48 x + 192 - 27 = 0

$3 x^{2} - 48 x + 192 - 27 = 0$

Step 2.1.2

Subtract

27

$27$ from

192

$192$ .

3 x^{2} - 48 x + 165 = 0

$3 x^{2} - 48 x + 165 = 0$

3 x^{2} - 48 x + 165 = 0

$3 x^{2} - 48 x + 165 = 0$

Step 2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

x = 5, 11

$x = 5, 11$

x = 5, 11

$x = 5, 11$

Step 3