Algebra Examples

$(2 w + 8) (2 w + 10) = 224$

Step 1

Simplify $(2 w + 8) (2 w + 10)$ .

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Step 1.1

Expand $(2 w + 8) (2 w + 10)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$2 w (2 w + 10) + 8 (2 w + 10) = 224$

Step 1.1.2

Apply the distributive property.

$2 w (2 w) + 2 w \cdot 10 + 8 (2 w + 10) = 224$

Step 1.1.3

Apply the distributive property.

$2 w (2 w) + 2 w \cdot 10 + 8 (2 w) + 8 \cdot 10 = 224$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 w \cdot w + 2 w \cdot 10 + 8 (2 w) + 8 \cdot 10 = 224$

Step 1.2.1.2

Multiply $w$ by $w$ by adding the exponents.

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Step 1.2.1.2.1

Move $w$ .

$2 \cdot 2 (w \cdot w) + 2 w \cdot 10 + 8 (2 w) + 8 \cdot 10 = 224$

Step 1.2.1.2.2

Multiply $w$ by $w$ .

$2 \cdot 2 w^{2} + 2 w \cdot 10 + 8 (2 w) + 8 \cdot 10 = 224$

Step 1.2.1.3

Multiply $2$ by $2$ .

$4 w^{2} + 2 w \cdot 10 + 8 (2 w) + 8 \cdot 10 = 224$

Step 1.2.1.4

Multiply $10$ by $2$ .

$4 w^{2} + 20 w + 8 (2 w) + 8 \cdot 10 = 224$

Step 1.2.1.5

Multiply $2$ by $8$ .

$4 w^{2} + 20 w + 16 w + 8 \cdot 10 = 224$

Step 1.2.1.6

Multiply $8$ by $10$ .

$4 w^{2} + 20 w + 16 w + 80 = 224$

Step 1.2.2

Add $20 w$ and $16 w$ .

$4 w^{2} + 36 w + 80 = 224$

Step 2

Subtract $224$ from both sides of the equation.

$4 w^{2} + 36 w + 80 - 224 = 0$

Step 3

Subtract $224$ from $80$ .

$4 w^{2} + 36 w - 144 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $4$ out of $4 w^{2} + 36 w - 144$ .

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Step 4.1.1

Factor $4$ out of $4 w^{2}$ .

$4 (w^{2}) + 36 w - 144 = 0$

Step 4.1.2

Factor $4$ out of $36 w$ .

$4 (w^{2}) + 4 (9 w) - 144 = 0$

Step 4.1.3

Factor $4$ out of $- 144$ .

$4 w^{2} + 4 (9 w) + 4 \cdot - 36 = 0$

Step 4.1.4

Factor $4$ out of $4 w^{2} + 4 (9 w)$ .

$4 (w^{2} + 9 w) + 4 \cdot - 36 = 0$

Step 4.1.5

Factor $4$ out of $4 (w^{2} + 9 w) + 4 \cdot - 36$ .

$4 (w^{2} + 9 w - 36) = 0$

Step 4.2

Factor.

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Step 4.2.1

Factor $w^{2} + 9 w - 36$ using the AC method.

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Step 4.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 36$ and whose sum is $9$ .

$- 3, 12$

Step 4.2.1.2

Write the factored form using these integers.

$4 ((w - 3) (w + 12)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$4 (w - 3) (w + 12) = 0$

Step 5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$w - 3 = 0$

$w + 12 = 0$

Step 6

Set $w - 3$ equal to $0$ and solve for $w$ .

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Step 6.1

Set $w - 3$ equal to $0$ .

$w - 3 = 0$

Step 6.2

Add $3$ to both sides of the equation.

$w = 3$

Step 7

Set $w + 12$ equal to $0$ and solve for $w$ .

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Step 7.1

Set $w + 12$ equal to $0$ .

$w + 12 = 0$

Step 7.2

Subtract $12$ from both sides of the equation.

$w = - 12$

Step 8

The final solution is all the values that make $4 (w - 3) (w + 12) = 0$ true.

$w = 3, - 12$