Basic Math Examples

${(u + 4)}^{2} = 2 u + 5 u - 12$

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.

$2 u + 5 u - 12 = {(u + 4)}^{2}$

Step 2

Add $2 u$ and $5 u$ .

$7 u - 12 = {(u + 4)}^{2}$

Step 3

Simplify ${(u + 4)}^{2}$ .

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Step 3.1

Rewrite ${(u + 4)}^{2}$ as $(u + 4) (u + 4)$ .

$7 u - 12 = (u + 4) (u + 4)$

Step 3.2

Expand $(u + 4) (u + 4)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$7 u - 12 = u (u + 4) + 4 (u + 4)$

Step 3.2.2

Apply the distributive property.

$7 u - 12 = u \cdot u + u \cdot 4 + 4 (u + 4)$

Step 3.2.3

Apply the distributive property.

$7 u - 12 = u \cdot u + u \cdot 4 + 4 u + 4 \cdot 4$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $u$ by $u$ .

$7 u - 12 = u^{2} + u \cdot 4 + 4 u + 4 \cdot 4$

Step 3.3.1.2

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

$7 u - 12 = u^{2} + 4 \cdot u + 4 u + 4 \cdot 4$

Step 3.3.1.3

Multiply $4$ by $4$ .

$7 u - 12 = u^{2} + 4 u + 4 u + 16$

Step 3.3.2

Add $4 u$ and $4 u$ .

$7 u - 12 = u^{2} + 8 u + 16$

Step 4

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

$u^{2} + 8 u + 16 = 7 u - 12$

Step 5

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

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Step 5.1

Subtract $7 u$ from both sides of the equation.

$u^{2} + 8 u + 16 - 7 u = - 12$

Step 5.2

Subtract $7 u$ from $8 u$ .

$u^{2} + u + 16 = - 12$

Step 6

Move all terms to the left side of the equation and simplify.

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Step 6.1

Add $12$ to both sides of the equation.

$u^{2} + u + 16 + 12 = 0$

Step 6.2

Add $16$ and $12$ .

$u^{2} + u + 28 = 0$

Step 7

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 8

Substitute the values $a = 1$ , $b = 1$ , and $c = 28$ into the quadratic formula and solve for $u$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot 28)}}{2 \cdot 1}$

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

One to any power is one.

$u = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 28}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot 28$ .

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 1 \pm \sqrt{1 - 4 \cdot 28}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $28$ .

$u = \frac{- 1 \pm \sqrt{1 - 112}}{2 \cdot 1}$

Step 9.1.3

Subtract $112$ from $1$ .

$u = \frac{- 1 \pm \sqrt{- 111}}{2 \cdot 1}$

Step 9.1.4

Rewrite $- 111$ as $- 1 (111)$ .

$u = \frac{- 1 \pm \sqrt{- 1 \cdot 111}}{2 \cdot 1}$

Step 9.1.5

Rewrite $\sqrt{- 1 (111)}$ as $\sqrt{- 1} \cdot \sqrt{111}$ .

$u = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{111}}{2 \cdot 1}$

Step 9.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{- 1 \pm i \sqrt{111}}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$u = \frac{- 1 \pm i \sqrt{111}}{2}$

Step 10

The final answer is the combination of both solutions.

$u = - \frac{1 - i \sqrt{111}}{2}, - \frac{1 + i \sqrt{111}}{2}$