Algebra Examples

$- 5 {(x + 1)}^{2} + 5$

Step 1

Write $- 5 {(x + 1)}^{2} + 5$ as an equation.

$y = - 5 {(x + 1)}^{2} + 5$

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 $x$ .

$0 = - 5 {(x + 1)}^{2} + 5$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$0 = - 5 {(x + 1)}^{2} + 5$

Step 2.2.2

Simplify $- 5 {(x + 1)}^{2} + 5$ .

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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 $(x + 1) (x + 1)$ .

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

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.

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

Step 2.2.2.1.2.2

Apply the distributive property.

$0 = - 5 (x \cdot x + x \cdot 1 + 1 (x + 1)) + 5$

Step 2.2.2.1.2.3

Apply the distributive property.

$0 = - 5 (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) + 5$

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 $x$ .

$0 = - 5 (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) + 5$

Step 2.2.2.1.3.1.2

Multiply $x$ by $1$ .

$0 = - 5 (x^{2} + x + 1 x + 1 \cdot 1) + 5$

Step 2.2.2.1.3.1.3

Multiply $x$ by $1$ .

$0 = - 5 (x^{2} + x + x + 1 \cdot 1) + 5$

Step 2.2.2.1.3.1.4

Multiply $1$ by $1$ .

$0 = - 5 (x^{2} + x + x + 1) + 5$

Step 2.2.2.1.3.2

Add $x$ and $x$ .

$0 = - 5 (x^{2} + 2 x + 1) + 5$

Step 2.2.2.1.4

Apply the distributive property.

$0 = - 5 x^{2} - 5 (2 x) - 5 \cdot 1 + 5$

Step 2.2.2.1.5

Simplify.

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Step 2.2.2.1.5.1

Multiply $2$ by $- 5$ .

$0 = - 5 x^{2} - 10 x - 5 \cdot 1 + 5$

Step 2.2.2.1.5.2

Multiply $- 5$ by $1$ .

$0 = - 5 x^{2} - 10 x - 5 + 5$

Step 2.2.2.2

Combine the opposite terms in $- 5 x^{2} - 10 x - 5 + 5$ .

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Step 2.2.2.2.1

Add $- 5$ and $5$ .

$0 = - 5 x^{2} - 10 x + 0$

Step 2.2.2.2.2

Add $- 5 x^{2} - 10 x$ and $0$ .

$0 = - 5 x^{2} - 10 x$

Step 2.2.3

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

$x = - 2, 0$

Step 2.3

x-intercept(s) in point form.

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

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 $y$ .

$y = - 5 {((0) + 1)}^{2} + 5$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = - 5 {(0 + 1)}^{2} + 5$

Step 3.2.2

Remove parentheses.

$y = - 5 {((0) + 1)}^{2} + 5$

Step 3.2.3

Simplify $- 5 {((0) + 1)}^{2} + 5$ .

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Add $0$ and $1$ .

$y = - 5 \cdot 1^{2} + 5$

Step 3.2.3.1.2

One to any power is one.

$y = - 5 \cdot 1 + 5$

Step 3.2.3.1.3

Multiply $- 5$ by $1$ .

$y = - 5 + 5$

Step 3.2.3.2

Add $- 5$ and $5$ .

$y = 0$

Step 3.3

y-intercept(s) in point form.

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

Step 4

List the intersections.

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

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

Step 5