Pre-Algebra Examples

${(u + 5)}^{2} = 2 u^{2} + 14 u + 13$

Step 1

Since $u$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 u^{2} + 14 u + 13 = {(u + 5)}^{2}$

Step 2

Simplify ${(u + 5)}^{2}$ .

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Step 2.1

Rewrite.

$2 u^{2} + 14 u + 13 = 0 + 0 + {(u + 5)}^{2}$

Step 2.2

Rewrite ${(u + 5)}^{2}$ as $(u + 5) (u + 5)$ .

$2 u^{2} + 14 u + 13 = (u + 5) (u + 5)$

Step 2.3

Expand $(u + 5) (u + 5)$ using the FOIL Method.

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Step 2.3.1

Apply the distributive property.

$2 u^{2} + 14 u + 13 = u (u + 5) + 5 (u + 5)$

Step 2.3.2

Apply the distributive property.

$2 u^{2} + 14 u + 13 = u \cdot u + u \cdot 5 + 5 (u + 5)$

Step 2.3.3

Apply the distributive property.

$2 u^{2} + 14 u + 13 = u \cdot u + u \cdot 5 + 5 u + 5 \cdot 5$

Step 2.4

Simplify and combine like terms.

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

Multiply $u$ by $u$ .

$2 u^{2} + 14 u + 13 = u^{2} + u \cdot 5 + 5 u + 5 \cdot 5$

Step 2.4.1.2

Move $5$ to the left of $u$ .

$2 u^{2} + 14 u + 13 = u^{2} + 5 \cdot u + 5 u + 5 \cdot 5$

Step 2.4.1.3

Multiply $5$ by $5$ .

$2 u^{2} + 14 u + 13 = u^{2} + 5 u + 5 u + 25$

Step 2.4.2

Add $5 u$ and $5 u$ .

$2 u^{2} + 14 u + 13 = u^{2} + 10 u + 25$

Step 3

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

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Step 3.1

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

$2 u^{2} + 14 u + 13 - u^{2} = 10 u + 25$

Step 3.2

Subtract $10 u$ from both sides of the equation.

$2 u^{2} + 14 u + 13 - u^{2} - 10 u = 25$

Step 3.3

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

$u^{2} + 14 u + 13 - 10 u = 25$

Step 3.4

Subtract $10 u$ from $14 u$ .

$u^{2} + 4 u + 13 = 25$

Step 4

Subtract $25$ from both sides of the equation.

$u^{2} + 4 u + 13 - 25 = 0$

Step 5

Subtract $25$ from $13$ .

$u^{2} + 4 u - 12 = 0$

Step 6

Factor $u^{2} + 4 u - 12$ using the AC method.

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Step 6.1

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 $- 12$ and whose sum is $4$ .

$- 2, 6$

Step 6.2

Write the factored form using these integers.

$(u - 2) (u + 6) = 0$

Step 7

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

$u - 2 = 0$

$u + 6 = 0$

Step 8

Set $u - 2$ equal to $0$ and solve for $u$ .

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Step 8.1

Set $u - 2$ equal to $0$ .

$u - 2 = 0$

Step 8.2

Add $2$ to both sides of the equation.

$u = 2$

Step 9

Set $u + 6$ equal to $0$ and solve for $u$ .

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Step 9.1

Set $u + 6$ equal to $0$ .

$u + 6 = 0$

Step 9.2

Subtract $6$ from both sides of the equation.

$u = - 6$

Step 10

The final solution is all the values that make $(u - 2) (u + 6) = 0$ true.

$u = 2, - 6$