Basic Math Examples

$\sqrt{c^{2} + 44 b} = b + 11$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{c^{2} + 44 b}}^{2} = {(b + 11)}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{c^{2} + 44 b}$ as ${(c^{2} + 44 b)}^{\frac{1}{2}}$ .

${({(c^{2} + 44 b)}^{\frac{1}{2}})}^{2} = {(b + 11)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(c^{2} + 44 b)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(c^{2} + 44 b)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(c^{2} + 44 b)}^{\frac{1}{2} \cdot 2} = {(b + 11)}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(c^{2} + 44 b)}^{\frac{1}{2} \cdot 2} = {(b + 11)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(c^{2} + 44 b)}^{1} = {(b + 11)}^{2}$

Step 2.2.1.2

Simplify.

$c^{2} + 44 b = {(b + 11)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(b + 11)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(b + 11)}^{2}$ as $(b + 11) (b + 11)$ .

$c^{2} + 44 b = (b + 11) (b + 11)$

Step 2.3.1.2

Expand $(b + 11) (b + 11)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$c^{2} + 44 b = b (b + 11) + 11 (b + 11)$

Step 2.3.1.2.2

Apply the distributive property.

$c^{2} + 44 b = b \cdot b + b \cdot 11 + 11 (b + 11)$

Step 2.3.1.2.3

Apply the distributive property.

$c^{2} + 44 b = b \cdot b + b \cdot 11 + 11 b + 11 \cdot 11$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $b$ by $b$ .

$c^{2} + 44 b = b^{2} + b \cdot 11 + 11 b + 11 \cdot 11$

Step 2.3.1.3.1.2

Move $11$ to the left of $b$ .

$c^{2} + 44 b = b^{2} + 11 \cdot b + 11 b + 11 \cdot 11$

Step 2.3.1.3.1.3

Multiply $11$ by $11$ .

$c^{2} + 44 b = b^{2} + 11 b + 11 b + 121$

Step 2.3.1.3.2

Add $11 b$ and $11 b$ .

$c^{2} + 44 b = b^{2} + 22 b + 121$

Step 3

Solve for $b$ .

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Step 3.1

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

$b^{2} + 22 b + 121 = c^{2} + 44 b$

Step 3.2

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

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Step 3.2.1

Subtract $44 b$ from both sides of the equation.

$b^{2} + 22 b + 121 - 44 b = c^{2}$

Step 3.2.2

Subtract $44 b$ from $22 b$ .

$b^{2} - 22 b + 121 = c^{2}$

Step 3.3

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

$b^{2} - 22 b + 121 - c^{2} = 0$

Step 3.4

Use the quadratic formula to find the solutions.

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

Step 3.5

Substitute the values $a = 1$ , $b = - 22$ , and $c = 121 - c^{2}$ into the quadratic formula and solve for $b$ .

$\frac{22 \pm \sqrt{{(- 22)}^{2} - 4 \cdot (1 \cdot (121 - c^{2}))}}{2 \cdot 1}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $- 22$ to the power of $2$ .

$b = \frac{22 \pm \sqrt{484 - 4 \cdot 1 \cdot (121 - c^{2})}}{2 \cdot 1}$

Step 3.6.1.2

Multiply $- 4$ by $1$ .

$b = \frac{22 \pm \sqrt{484 - 4 \cdot (121 - c^{2})}}{2 \cdot 1}$

Step 3.6.1.3

Apply the distributive property.

$b = \frac{22 \pm \sqrt{484 - 4 \cdot 121 - 4 (- c^{2})}}{2 \cdot 1}$

Step 3.6.1.4

Multiply $- 4$ by $121$ .

$b = \frac{22 \pm \sqrt{484 - 484 - 4 (- c^{2})}}{2 \cdot 1}$

Step 3.6.1.5

Multiply $- 1$ by $- 4$ .

$b = \frac{22 \pm \sqrt{484 - 484 + 4 c^{2}}}{2 \cdot 1}$

Step 3.6.1.6

Subtract $484$ from $484$ .

$b = \frac{22 \pm \sqrt{0 + 4 c^{2}}}{2 \cdot 1}$

Step 3.6.1.7

Add $0$ and $4 c^{2}$ .

$b = \frac{22 \pm \sqrt{4 c^{2}}}{2 \cdot 1}$

Step 3.6.1.8

Rewrite $4 c^{2}$ as ${(2 c)}^{2}$ .

$b = \frac{22 \pm \sqrt{{(2 c)}^{2}}}{2 \cdot 1}$

Step 3.6.1.9

Pull terms out from under the radical, assuming positive real numbers.

$b = \frac{22 \pm 2 c}{2 \cdot 1}$

Step 3.6.2

Multiply $2$ by $1$ .

$b = \frac{22 \pm 2 c}{2}$

Step 3.6.3

Simplify $\frac{22 \pm 2 c}{2}$ .

$b = 11 \pm c$

Step 3.7

The final answer is the combination of both solutions.

$b = 11 + c$

$b = 11 - c$

$b = 11 + c$

$b = 11 - c$