Precalculus Examples

$f (x) = 16 - {(x - 1)}^{2}$

Step 1

Set $16 - {(x - 1)}^{2}$ equal to $0$ .

$16 - {(x - 1)}^{2} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Simplify $16 - {(x - 1)}^{2}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$16 - ((x - 1) (x - 1)) = 0$

Step 2.1.1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 2.1.1.2.1

Apply the distributive property.

$16 - (x (x - 1) - 1 (x - 1)) = 0$

Step 2.1.1.2.2

Apply the distributive property.

$16 - (x \cdot x + x \cdot - 1 - 1 (x - 1)) = 0$

Step 2.1.1.2.3

Apply the distributive property.

$16 - (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) = 0$

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 $x$ by $x$ .

$16 - (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) = 0$

Step 2.1.1.3.1.2

Move $- 1$ to the left of $x$ .

$16 - (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) = 0$

Step 2.1.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

$16 - (x^{2} - x - 1 x - 1 \cdot - 1) = 0$

Step 2.1.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

$16 - (x^{2} - x - x - 1 \cdot - 1) = 0$

Step 2.1.1.3.1.5

Multiply $- 1$ by $- 1$ .

$16 - (x^{2} - x - x + 1) = 0$

Step 2.1.1.3.2

Subtract $x$ from $- x$ .

$16 - (x^{2} - 2 x + 1) = 0$

Step 2.1.1.4

Apply the distributive property.

$16 - x^{2} - (- 2 x) - 1 \cdot 1 = 0$

Step 2.1.1.5

Simplify.

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Step 2.1.1.5.1

Multiply $- 2$ by $- 1$ .

$16 - x^{2} + 2 x - 1 \cdot 1 = 0$

Step 2.1.1.5.2

Multiply $- 1$ by $1$ .

$16 - x^{2} + 2 x - 1 = 0$

Step 2.1.2

Subtract $1$ from $16$ .

$- x^{2} + 2 x + 15 = 0$

Step 2.2

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

$x = - 3, 5$

Step 3