Basic Math Examples

$\sqrt{c^{2} + 5 c + 7} = c + 4$

Step 1

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

${\sqrt{c^{2} + 5 c + 7}}^{2} = {(c + 4)}^{2}$

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 $\sqrt{c^{2} + 5 c + 7}$ as ${(c^{2} + 5 c + 7)}^{\frac{1}{2}}$ .

${({(c^{2} + 5 c + 7)}^{\frac{1}{2}})}^{2} = {(c + 4)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(c^{2} + 5 c + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(c^{2} + 5 c + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(c^{2} + 5 c + 7)}^{\frac{1}{2} \cdot 2} = {(c + 4)}^{2}$

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.

${(c^{2} + 5 c + 7)}^{\frac{1}{2} \cdot 2} = {(c + 4)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(c^{2} + 5 c + 7)}^{1} = {(c + 4)}^{2}$

Step 2.2.1.2

Simplify.

$c^{2} + 5 c + 7 = {(c + 4)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

$c^{2} + 5 c + 7 = (c + 4) (c + 4)$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Apply the distributive property.

$c^{2} + 5 c + 7 = c (c + 4) + 4 (c + 4)$

Step 2.3.1.2.2

Apply the distributive property.

$c^{2} + 5 c + 7 = c \cdot c + c \cdot 4 + 4 (c + 4)$

Step 2.3.1.2.3

Apply the distributive property.

$c^{2} + 5 c + 7 = c \cdot c + c \cdot 4 + 4 c + 4 \cdot 4$

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 $c$ by $c$ .

$c^{2} + 5 c + 7 = c^{2} + c \cdot 4 + 4 c + 4 \cdot 4$

Step 2.3.1.3.1.2

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

$c^{2} + 5 c + 7 = c^{2} + 4 \cdot c + 4 c + 4 \cdot 4$

Step 2.3.1.3.1.3

Multiply $4$ by $4$ .

$c^{2} + 5 c + 7 = c^{2} + 4 c + 4 c + 16$

Step 2.3.1.3.2

Add $4 c$ and $4 c$ .

$c^{2} + 5 c + 7 = c^{2} + 8 c + 16$

Step 3

Solve for $c$ .

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Step 3.1

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

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Step 3.1.1

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

$c^{2} + 5 c + 7 - c^{2} = 8 c + 16$

Step 3.1.2

Subtract $8 c$ from both sides of the equation.

$c^{2} + 5 c + 7 - c^{2} - 8 c = 16$

Step 3.1.3

Combine the opposite terms in $c^{2} + 5 c + 7 - c^{2} - 8 c$ .

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Step 3.1.3.1

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

$5 c + 7 + 0 - 8 c = 16$

Step 3.1.3.2

Add $5 c + 7$ and $0$ .

$5 c + 7 - 8 c = 16$

Step 3.1.4

Subtract $8 c$ from $5 c$ .

$- 3 c + 7 = 16$

Step 3.2

Move all terms not containing $c$ to the right side of the equation.

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Step 3.2.1

Subtract $7$ from both sides of the equation.

$- 3 c = 16 - 7$

Step 3.2.2

Subtract $7$ from $16$ .

$- 3 c = 9$

Step 3.3

Divide each term in $- 3 c = 9$ by $- 3$ and simplify.

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Step 3.3.1

Divide each term in $- 3 c = 9$ by $- 3$ .

$\frac{- 3 c}{- 3} = \frac{9}{- 3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 3$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 3 c}{- 3} = \frac{9}{- 3}$

Step 3.3.2.1.2

Divide $c$ by $1$ .

$c = \frac{9}{- 3}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $9$ by $- 3$ .

$c = - 3$