Pre-Algebra Examples

$2 v^{2} + 20 v + 51 = {(v + 6)}^{2}$

Step 1

Simplify ${(v + 6)}^{2}$ .

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Step 1.1

Rewrite.

$2 v^{2} + 20 v + 51 = 0 + 0 + {(v + 6)}^{2}$

Step 1.2

Rewrite ${(v + 6)}^{2}$ as $(v + 6) (v + 6)$ .

$2 v^{2} + 20 v + 51 = (v + 6) (v + 6)$

Step 1.3

Expand $(v + 6) (v + 6)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$2 v^{2} + 20 v + 51 = v (v + 6) + 6 (v + 6)$

Step 1.3.2

Apply the distributive property.

$2 v^{2} + 20 v + 51 = v \cdot v + v \cdot 6 + 6 (v + 6)$

Step 1.3.3

Apply the distributive property.

$2 v^{2} + 20 v + 51 = v \cdot v + v \cdot 6 + 6 v + 6 \cdot 6$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $v$ by $v$ .

$2 v^{2} + 20 v + 51 = v^{2} + v \cdot 6 + 6 v + 6 \cdot 6$

Step 1.4.1.2

Move $6$ to the left of $v$ .

$2 v^{2} + 20 v + 51 = v^{2} + 6 \cdot v + 6 v + 6 \cdot 6$

Step 1.4.1.3

Multiply $6$ by $6$ .

$2 v^{2} + 20 v + 51 = v^{2} + 6 v + 6 v + 36$

Step 1.4.2

Add $6 v$ and $6 v$ .

$2 v^{2} + 20 v + 51 = v^{2} + 12 v + 36$

Step 2

Move all terms containing $v$ to the left side of the equation.

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Step 2.1

Subtract $v^{2}$ from both sides of the equation.

$2 v^{2} + 20 v + 51 - v^{2} = 12 v + 36$

Step 2.2

Subtract $12 v$ from both sides of the equation.

$2 v^{2} + 20 v + 51 - v^{2} - 12 v = 36$

Step 2.3

Subtract $v^{2}$ from $2 v^{2}$ .

$v^{2} + 20 v + 51 - 12 v = 36$

Step 2.4

Subtract $12 v$ from $20 v$ .

$v^{2} + 8 v + 51 = 36$

Step 3

Subtract $36$ from both sides of the equation.

$v^{2} + 8 v + 51 - 36 = 0$

Step 4

Subtract $36$ from $51$ .

$v^{2} + 8 v + 15 = 0$

Step 5

Factor $v^{2} + 8 v + 15$ using the AC method.

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Step 5.1

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 $15$ and whose sum is $8$ .

$3, 5$

Step 5.2

Write the factored form using these integers.

$(v + 3) (v + 5) = 0$

Step 6

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

$v + 3 = 0$

$v + 5 = 0$

Step 7

Set $v + 3$ equal to $0$ and solve for $v$ .

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Step 7.1

Set $v + 3$ equal to $0$ .

$v + 3 = 0$

Step 7.2

Subtract $3$ from both sides of the equation.

$v = - 3$

Step 8

Set $v + 5$ equal to $0$ and solve for $v$ .

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Step 8.1

Set $v + 5$ equal to $0$ .

$v + 5 = 0$

Step 8.2

Subtract $5$ from both sides of the equation.

$v = - 5$

Step 9

The final solution is all the values that make $(v + 3) (v + 5) = 0$ true.

$v = - 3, - 5$