Basic Math Examples

${(u + 5)}^{2} = 2 u - u + 55$

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.

$2 u - u + 55 = {(u + 5)}^{2}$

Step 2

Subtract $u$ from $2 u$ .

$u + 55 = {(u + 5)}^{2}$

Step 3

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

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Step 3.1

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

$u + 55 = (u + 5) (u + 5)$

Step 3.2

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

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Step 3.2.1

Apply the distributive property.

$u + 55 = u (u + 5) + 5 (u + 5)$

Step 3.2.2

Apply the distributive property.

$u + 55 = u \cdot u + u \cdot 5 + 5 (u + 5)$

Step 3.2.3

Apply the distributive property.

$u + 55 = u \cdot u + u \cdot 5 + 5 u + 5 \cdot 5$

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 $u$ .

$u + 55 = u^{2} + u \cdot 5 + 5 u + 5 \cdot 5$

Step 3.3.1.2

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

$u + 55 = u^{2} + 5 \cdot u + 5 u + 5 \cdot 5$

Step 3.3.1.3

Multiply $5$ by $5$ .

$u + 55 = u^{2} + 5 u + 5 u + 25$

Step 3.3.2

Add $5 u$ and $5 u$ .

$u + 55 = u^{2} + 10 u + 25$

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.

$u^{2} + 10 u + 25 = u + 55$

Step 5

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

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Step 5.1

Subtract $u$ from both sides of the equation.

$u^{2} + 10 u + 25 - u = 55$

Step 5.2

Subtract $u$ from $10 u$ .

$u^{2} + 9 u + 25 = 55$

Step 6

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

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Step 6.1

Subtract $55$ from both sides of the equation.

$u^{2} + 9 u + 25 - 55 = 0$

Step 6.2

Subtract $55$ from $25$ .

$u^{2} + 9 u - 30 = 0$

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 $a = 1$ , $b = 9$ , and $c = - 30$ into the quadratic formula and solve for $u$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (1 \cdot - 30)}}{2 \cdot 1}$

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $9$ to the power of $2$ .

$u = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1 \cdot - 30}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot - 30$ .

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 9 \pm \sqrt{81 - 4 \cdot - 30}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $- 30$ .

$u = \frac{- 9 \pm \sqrt{81 + 120}}{2 \cdot 1}$

Step 9.1.3

Add $81$ and $120$ .

$u = \frac{- 9 \pm \sqrt{201}}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$u = \frac{- 9 \pm \sqrt{201}}{2}$

Step 10

The final answer is the combination of both solutions.

$u = - \frac{9 - \sqrt{201}}{2}, - \frac{9 + \sqrt{201}}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$u = - \frac{9 - \sqrt{201}}{2}, - \frac{9 + \sqrt{201}}{2}$

Decimal Form:

$u = 2.58872343 \dots, - 11.58872343 \dots$