Algebra Examples

$- {(x - 1)}^{2} + 9$

Step 1

Write $- {(x - 1)}^{2} + 9$ as an equation.

$y = - {(x - 1)}^{2} + 9$

Step 2

Find the x-intercepts.

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Step 2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - {(x - 1)}^{2} + 9$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$0 = - {(x - 1)}^{2} + 9$

Step 2.2.2

Simplify $- {(x - 1)}^{2} + 9$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

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

$0 = - ((x - 1) (x - 1)) + 9$

Step 2.2.2.1.2

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

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Step 2.2.2.1.2.1

Apply the distributive property.

$0 = - (x (x - 1) - 1 (x - 1)) + 9$

Step 2.2.2.1.2.2

Apply the distributive property.

$0 = - (x \cdot x + x \cdot - 1 - 1 (x - 1)) + 9$

Step 2.2.2.1.2.3

Apply the distributive property.

$0 = - (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) + 9$

Step 2.2.2.1.3

Simplify and combine like terms.

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Step 2.2.2.1.3.1

Simplify each term.

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Step 2.2.2.1.3.1.1

Multiply $x$ by $x$ .

$0 = - (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) + 9$

Step 2.2.2.1.3.1.2

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

$0 = - (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) + 9$

Step 2.2.2.1.3.1.3

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

$0 = - (x^{2} - x - 1 x - 1 \cdot - 1) + 9$

Step 2.2.2.1.3.1.4

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

$0 = - (x^{2} - x - x - 1 \cdot - 1) + 9$

Step 2.2.2.1.3.1.5

Multiply $- 1$ by $- 1$ .

$0 = - (x^{2} - x - x + 1) + 9$

Step 2.2.2.1.3.2

Subtract $x$ from $- x$ .

$0 = - (x^{2} - 2 x + 1) + 9$

Step 2.2.2.1.4

Apply the distributive property.

$0 = - x^{2} - (- 2 x) - 1 \cdot 1 + 9$

Step 2.2.2.1.5

Simplify.

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Step 2.2.2.1.5.1

Multiply $- 2$ by $- 1$ .

$0 = - x^{2} + 2 x - 1 \cdot 1 + 9$

Step 2.2.2.1.5.2

Multiply $- 1$ by $1$ .

$0 = - x^{2} + 2 x - 1 + 9$

Step 2.2.2.2

Add $- 1$ and $9$ .

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

Step 2.2.3

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

$x = - 2, 4$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(- 2, 0), (4, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - {((0) - 1)}^{2} + 9$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = - {(0 - 1)}^{2} + 9$

Step 3.2.2

Remove parentheses.

$y = - {((0) - 1)}^{2} + 9$

Step 3.2.3

Simplify $- {((0) - 1)}^{2} + 9$ .

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Subtract $1$ from $0$ .

$y = - {(- 1)}^{2} + 9$

Step 3.2.3.1.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 3.2.3.1.2.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 3.2.3.1.2.1.1

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

$y = {(- 1)}^{1} {(- 1)}^{2} + 9$

Step 3.2.3.1.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = {(- 1)}^{1 + 2} + 9$

Step 3.2.3.1.2.2

Add $1$ and $2$ .

$y = {(- 1)}^{3} + 9$

Step 3.2.3.1.3

Raise $- 1$ to the power of $3$ .

$y = - 1 + 9$

Step 3.2.3.2

Add $- 1$ and $9$ .

$y = 8$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, 8)$

Step 4

List the intersections.

x-intercept(s): $(- 2, 0), (4, 0)$

y-intercept(s): $(0, 8)$

Step 5