Calculus Examples

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

Step 1

Simplify $(x - 4) {(x + 2)}^{2}$ .

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Step 1.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$0 = (x - 4) ((x + 2) (x + 2))$

Step 1.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$0 = (x - 4) (x (x + 2) + 2 (x + 2))$

Step 1.2.2

Apply the distributive property.

$0 = (x - 4) (x \cdot x + x \cdot 2 + 2 (x + 2))$

Step 1.2.3

Apply the distributive property.

$0 = (x - 4) (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2)$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.3.1.2

Move $2$ to the left of $x$ .

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

Step 1.3.1.3

Multiply $2$ by $2$ .

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

Step 1.3.2

Add $2 x$ and $2 x$ .

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

Step 1.4

Expand $(x - 4) (x^{2} + 4 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.5

Simplify terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.5.1.1.1

Multiply $x$ by $x^{2}$ .

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Step 1.5.1.1.1.1

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

$0 = x^{1} x^{2} + x (4 x) + x \cdot 4 - 4 x^{2} - 4 (4 x) - 4 \cdot 4$

Step 1.5.1.1.1.2

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

$0 = x^{1 + 2} + x (4 x) + x \cdot 4 - 4 x^{2} - 4 (4 x) - 4 \cdot 4$

Step 1.5.1.1.2

Add $1$ and $2$ .

$0 = x^{3} + x (4 x) + x \cdot 4 - 4 x^{2} - 4 (4 x) - 4 \cdot 4$

Step 1.5.1.2

Rewrite using the commutative property of multiplication.

$0 = x^{3} + 4 x \cdot x + x \cdot 4 - 4 x^{2} - 4 (4 x) - 4 \cdot 4$

Step 1.5.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 1.5.1.3.1

Move $x$ .

$0 = x^{3} + 4 (x \cdot x) + x \cdot 4 - 4 x^{2} - 4 (4 x) - 4 \cdot 4$

Step 1.5.1.3.2

Multiply $x$ by $x$ .

$0 = x^{3} + 4 x^{2} + x \cdot 4 - 4 x^{2} - 4 (4 x) - 4 \cdot 4$

Step 1.5.1.4

Move $4$ to the left of $x$ .

$0 = x^{3} + 4 x^{2} + 4 \cdot x - 4 x^{2} - 4 (4 x) - 4 \cdot 4$

Step 1.5.1.5

Multiply $4$ by $- 4$ .

$0 = x^{3} + 4 x^{2} + 4 x - 4 x^{2} - 16 x - 4 \cdot 4$

Step 1.5.1.6

Multiply $- 4$ by $4$ .

$0 = x^{3} + 4 x^{2} + 4 x - 4 x^{2} - 16 x - 16$

Step 1.5.2

Simplify by adding terms.

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Step 1.5.2.1

Combine the opposite terms in $x^{3} + 4 x^{2} + 4 x - 4 x^{2} - 16 x - 16$ .

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Step 1.5.2.1.1

Subtract $4 x^{2}$ from $4 x^{2}$ .

$0 = x^{3} + 4 x + 0 - 16 x - 16$

Step 1.5.2.1.2

Add $x^{3} + 4 x$ and $0$ .

$0 = x^{3} + 4 x - 16 x - 16$

Step 1.5.2.2

Subtract $16 x$ from $4 x$ .

$0 = x^{3} - 12 x - 16$

Step 2

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

$x = - 2, 4$

Step 3