Finite Math Examples

$\sqrt{4 x + 7} = \sqrt{x + 1} - \sqrt{x + 6}$

Step 1

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

${\sqrt{4 x + 7}}^{2} = {(\sqrt{x + 1} - \sqrt{x + 6})}^{2}$

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 $\sqrt{4 x + 7}$ as ${(4 x + 7)}^{\frac{1}{2}}$ .

${({(4 x + 7)}^{\frac{1}{2}})}^{2} = {(\sqrt{x + 1} - \sqrt{x + 6})}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(4 x + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(4 x + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(4 x + 7)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 1} - \sqrt{x + 6})}^{2}$

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.

${(4 x + 7)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 1} - \sqrt{x + 6})}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(4 x + 7)}^{1} = {(\sqrt{x + 1} - \sqrt{x + 6})}^{2}$

Step 2.2.1.2

Simplify.

$4 x + 7 = {(\sqrt{x + 1} - \sqrt{x + 6})}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(\sqrt{x + 1} - \sqrt{x + 6})}^{2}$ .

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Step 2.3.1.1

Rewrite ${(\sqrt{x + 1} - \sqrt{x + 6})}^{2}$ as $(\sqrt{x + 1} - \sqrt{x + 6}) (\sqrt{x + 1} - \sqrt{x + 6})$ .

$4 x + 7 = (\sqrt{x + 1} - \sqrt{x + 6}) (\sqrt{x + 1} - \sqrt{x + 6})$

Step 2.3.1.2

Expand $(\sqrt{x + 1} - \sqrt{x + 6}) (\sqrt{x + 1} - \sqrt{x + 6})$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$4 x + 7 = \sqrt{x + 1} (\sqrt{x + 1} - \sqrt{x + 6}) - \sqrt{x + 6} (\sqrt{x + 1} - \sqrt{x + 6})$

Step 2.3.1.2.2

Apply the distributive property.

$4 x + 7 = \sqrt{x + 1} \sqrt{x + 1} + \sqrt{x + 1} (- \sqrt{x + 6}) - \sqrt{x + 6} (\sqrt{x + 1} - \sqrt{x + 6})$

Step 2.3.1.2.3

Apply the distributive property.

$4 x + 7 = \sqrt{x + 1} \sqrt{x + 1} + \sqrt{x + 1} (- \sqrt{x + 6}) - \sqrt{x + 6} \sqrt{x + 1} - \sqrt{x + 6} (- \sqrt{x + 6})$

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 $\sqrt{x + 1} \sqrt{x + 1}$ .

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Step 2.3.1.3.1.1.1

Raise $\sqrt{x + 1}$ to the power of $1$ .

$4 x + 7 = {\sqrt{x + 1}}^{1} \sqrt{x + 1} + \sqrt{x + 1} (- \sqrt{x + 6}) - \sqrt{x + 6} \sqrt{x + 1} - \sqrt{x + 6} (- \sqrt{x + 6})$

Step 2.3.1.3.1.1.2

Raise $\sqrt{x + 1}$ to the power of $1$ .

$4 x + 7 = {\sqrt{x + 1}}^{1} {\sqrt{x + 1}}^{1} + \sqrt{x + 1} (- \sqrt{x + 6}) - \sqrt{x + 6} \sqrt{x + 1} - \sqrt{x + 6} (- \sqrt{x + 6})$

Step 2.3.1.3.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 x + 7 = {\sqrt{x + 1}}^{1 + 1} + \sqrt{x + 1} (- \sqrt{x + 6}) - \sqrt{x + 6} \sqrt{x + 1} - \sqrt{x + 6} (- \sqrt{x + 6})$

Step 2.3.1.3.1.1.4

Add $1$ and $1$ .

$4 x + 7 = {\sqrt{x + 1}}^{2} + \sqrt{x + 1} (- \sqrt{x + 6}) - \sqrt{x + 6} \sqrt{x + 1} - \sqrt{x + 6} (- \sqrt{x + 6})$

Step 2.3.1.3.1.2

Rewrite ${\sqrt{x + 1}}^{2}$ as $x + 1$ .

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Step 2.3.1.3.1.2.1

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

$4 x + 7 = {({(x + 1)}^{\frac{1}{2}})}^{2} + \sqrt{x + 1} (- \sqrt{x + 6}) - \sqrt{x + 6} \sqrt{x + 1} - \sqrt{x + 6} (- \sqrt{x + 6})$

Step 2.3.1.3.1.2.2

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

$4 x + 7 = {(x + 1)}^{\frac{1}{2} \cdot 2} + \sqrt{x + 1} (- \sqrt{x + 6}) - \sqrt{x + 6} \sqrt{x + 1} - \sqrt{x + 6} (- \sqrt{x + 6})$

Step 2.3.1.3.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$4 x + 7 = {(x + 1)}^{\frac{2}{2}} + \sqrt{x + 1} (- \sqrt{x + 6}) - \sqrt{x + 6} \sqrt{x + 1} - \sqrt{x + 6} (- \sqrt{x + 6})$

Step 2.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.2.4.1

Cancel the common factor.

$4 x + 7 = {(x + 1)}^{\frac{2}{2}} + \sqrt{x + 1} (- \sqrt{x + 6}) - \sqrt{x + 6} \sqrt{x + 1} - \sqrt{x + 6} (- \sqrt{x + 6})$

Step 2.3.1.3.1.2.4.2

Rewrite the expression.

$4 x + 7 = {(x + 1)}^{1} + \sqrt{x + 1} (- \sqrt{x + 6}) - \sqrt{x + 6} \sqrt{x + 1} - \sqrt{x + 6} (- \sqrt{x + 6})$

Step 2.3.1.3.1.2.5

Simplify.

$4 x + 7 = x + 1 + \sqrt{x + 1} (- \sqrt{x + 6}) - \sqrt{x + 6} \sqrt{x + 1} - \sqrt{x + 6} (- \sqrt{x + 6})$

Step 2.3.1.3.1.3

Rewrite using the commutative property of multiplication.

$4 x + 7 = x + 1 - \sqrt{x + 1} \sqrt{x + 6} - \sqrt{x + 6} \sqrt{x + 1} - \sqrt{x + 6} (- \sqrt{x + 6})$

Step 2.3.1.3.1.4

Combine using the product rule for radicals.

$4 x + 7 = x + 1 - \sqrt{(x + 6) (x + 1)} - \sqrt{x + 6} \sqrt{x + 1} - \sqrt{x + 6} (- \sqrt{x + 6})$

Step 2.3.1.3.1.5

Combine using the product rule for radicals.

$4 x + 7 = x + 1 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} - \sqrt{x + 6} (- \sqrt{x + 6})$

Step 2.3.1.3.1.6

Multiply $- \sqrt{x + 6} (- \sqrt{x + 6})$ .

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Step 2.3.1.3.1.6.1

Multiply $- 1$ by $- 1$ .

$4 x + 7 = x + 1 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} + 1 \sqrt{x + 6} \sqrt{x + 6}$

Step 2.3.1.3.1.6.2

Multiply $\sqrt{x + 6}$ by $1$ .

$4 x + 7 = x + 1 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} + \sqrt{x + 6} \sqrt{x + 6}$

Step 2.3.1.3.1.6.3

Raise $\sqrt{x + 6}$ to the power of $1$ .

$4 x + 7 = x + 1 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} + {\sqrt{x + 6}}^{1} \sqrt{x + 6}$

Step 2.3.1.3.1.6.4

Raise $\sqrt{x + 6}$ to the power of $1$ .

$4 x + 7 = x + 1 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} + {\sqrt{x + 6}}^{1} {\sqrt{x + 6}}^{1}$

Step 2.3.1.3.1.6.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 x + 7 = x + 1 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} + {\sqrt{x + 6}}^{1 + 1}$

Step 2.3.1.3.1.6.6

Add $1$ and $1$ .

$4 x + 7 = x + 1 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} + {\sqrt{x + 6}}^{2}$

Step 2.3.1.3.1.7

Rewrite ${\sqrt{x + 6}}^{2}$ as $x + 6$ .

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Step 2.3.1.3.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 6}$ as ${(x + 6)}^{\frac{1}{2}}$ .

$4 x + 7 = x + 1 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} + {({(x + 6)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.3.1.7.2

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

$4 x + 7 = x + 1 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} + {(x + 6)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3.1.7.3

Combine $\frac{1}{2}$ and $2$ .

$4 x + 7 = x + 1 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} + {(x + 6)}^{\frac{2}{2}}$

Step 2.3.1.3.1.7.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.7.4.1

Cancel the common factor.

$4 x + 7 = x + 1 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} + {(x + 6)}^{\frac{2}{2}}$

Step 2.3.1.3.1.7.4.2

Rewrite the expression.

$4 x + 7 = x + 1 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} + {(x + 6)}^{1}$

Step 2.3.1.3.1.7.5

Simplify.

$4 x + 7 = x + 1 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} + x + 6$

Step 2.3.1.3.2

Add $x$ and $x$ .

$4 x + 7 = 2 x + 1 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} + 6$

Step 2.3.1.3.3

Add $1$ and $6$ .

$4 x + 7 = 2 x + 7 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)}$

Step 3

Solve for $- \sqrt{(x + 6) (x + 1)}$ .

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Step 3.1

Rewrite the equation as $2 x + 7 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} = 4 x + 7$ .

$2 x + 7 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} = 4 x + 7$

Step 3.2

Move all terms not containing $- \sqrt{(x + 6) (x + 1)}$ to the right side of the equation.

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Step 3.2.1

Subtract $2 x$ from both sides of the equation.

$7 - \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} = 4 x + 7 - 2 x$

Step 3.2.2

Subtract $7$ from both sides of the equation.

$- \sqrt{(x + 6) (x + 1)} - \sqrt{(x + 1) (x + 6)} = 4 x + 7 - 2 x - 7$

Step 3.2.3

Add $\sqrt{(x + 1) (x + 6)}$ to both sides of the equation.

$- \sqrt{(x + 6) (x + 1)} = 4 x + 7 - 2 x - 7 + \sqrt{(x + 1) (x + 6)}$

Step 3.2.4

Combine the opposite terms in $4 x + 7 - 2 x - 7 + \sqrt{(x + 1) (x + 6)}$ .

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Step 3.2.4.1

Subtract $7$ from $7$ .

$- \sqrt{(x + 6) (x + 1)} = 4 x - 2 x + 0 + \sqrt{(x + 1) (x + 6)}$

Step 3.2.4.2

Add $4 x - 2 x$ and $0$ .

$- \sqrt{(x + 6) (x + 1)} = 4 x - 2 x + \sqrt{(x + 1) (x + 6)}$

Step 3.2.5

Subtract $2 x$ from $4 x$ .

$- \sqrt{(x + 6) (x + 1)} = 2 x + \sqrt{(x + 1) (x + 6)}$

Step 4

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

${(- \sqrt{(x + 6) (x + 1)})}^{2} = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x + 6) (x + 1)}$ as ${((x + 6) (x + 1))}^{\frac{1}{2}}$ .

${(- {((x + 6) (x + 1))}^{\frac{1}{2}})}^{2} = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${(- {((x + 6) (x + 1))}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Expand $(x + 6) (x + 1)$ using the FOIL Method.

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Step 5.2.1.1.1

Apply the distributive property.

${(- {(x (x + 1) + 6 (x + 1))}^{\frac{1}{2}})}^{2} = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5.2.1.1.2

Apply the distributive property.

${(- {(x \cdot x + x \cdot 1 + 6 (x + 1))}^{\frac{1}{2}})}^{2} = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5.2.1.1.3

Apply the distributive property.

${(- {(x \cdot x + x \cdot 1 + 6 x + 6 \cdot 1)}^{\frac{1}{2}})}^{2} = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5.2.1.2

Simplify and combine like terms.

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Step 5.2.1.2.1

Simplify each term.

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Step 5.2.1.2.1.1

Multiply $x$ by $x$ .

${(- {(x^{2} + x \cdot 1 + 6 x + 6 \cdot 1)}^{\frac{1}{2}})}^{2} = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5.2.1.2.1.2

Multiply $x$ by $1$ .

${(- {(x^{2} + x + 6 x + 6 \cdot 1)}^{\frac{1}{2}})}^{2} = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5.2.1.2.1.3

Multiply $6$ by $1$ .

${(- {(x^{2} + x + 6 x + 6)}^{\frac{1}{2}})}^{2} = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5.2.1.2.2

Add $x$ and $6 x$ .

${(- {(x^{2} + 7 x + 6)}^{\frac{1}{2}})}^{2} = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5.2.1.3

Apply the product rule to $- {(x^{2} + 7 x + 6)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(x^{2} + 7 x + 6)}^{\frac{1}{2}})}^{2} = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5.2.1.4

Raise $- 1$ to the power of $2$ .

$1 {({(x^{2} + 7 x + 6)}^{\frac{1}{2}})}^{2} = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5.2.1.5

Multiply ${({(x^{2} + 7 x + 6)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(x^{2} + 7 x + 6)}^{\frac{1}{2}})}^{2} = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5.2.1.6

Multiply the exponents in ${({(x^{2} + 7 x + 6)}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.6.1

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

${(x^{2} + 7 x + 6)}^{\frac{1}{2} \cdot 2} = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5.2.1.6.2

Cancel the common factor of $2$ .

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Step 5.2.1.6.2.1

Cancel the common factor.

${(x^{2} + 7 x + 6)}^{\frac{1}{2} \cdot 2} = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5.2.1.6.2.2

Rewrite the expression.

${(x^{2} + 7 x + 6)}^{1} = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5.2.1.7

Simplify.

$x^{2} + 7 x + 6 = {(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$ .

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Step 5.3.1.1

Rewrite ${(2 x + \sqrt{(x + 1) (x + 6)})}^{2}$ as $(2 x + \sqrt{(x + 1) (x + 6)}) (2 x + \sqrt{(x + 1) (x + 6)})$ .

$x^{2} + 7 x + 6 = (2 x + \sqrt{(x + 1) (x + 6)}) (2 x + \sqrt{(x + 1) (x + 6)})$

Step 5.3.1.2

Expand $(2 x + \sqrt{(x + 1) (x + 6)}) (2 x + \sqrt{(x + 1) (x + 6)})$ using the FOIL Method.

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Step 5.3.1.2.1

Apply the distributive property.

$x^{2} + 7 x + 6 = 2 x (2 x + \sqrt{(x + 1) (x + 6)}) + \sqrt{(x + 1) (x + 6)} (2 x + \sqrt{(x + 1) (x + 6)})$

Step 5.3.1.2.2

Apply the distributive property.

$x^{2} + 7 x + 6 = 2 x (2 x) + 2 x \sqrt{(x + 1) (x + 6)} + \sqrt{(x + 1) (x + 6)} (2 x + \sqrt{(x + 1) (x + 6)})$

Step 5.3.1.2.3

Apply the distributive property.

$x^{2} + 7 x + 6 = 2 x (2 x) + 2 x \sqrt{(x + 1) (x + 6)} + \sqrt{(x + 1) (x + 6)} (2 x) + \sqrt{(x + 1) (x + 6)} \sqrt{(x + 1) (x + 6)}$

Step 5.3.1.3

Simplify and combine like terms.

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Step 5.3.1.3.1

Simplify each term.

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Step 5.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$x^{2} + 7 x + 6 = 2 \cdot 2 x \cdot x + 2 x \sqrt{(x + 1) (x + 6)} + \sqrt{(x + 1) (x + 6)} (2 x) + \sqrt{(x + 1) (x + 6)} \sqrt{(x + 1) (x + 6)}$

Step 5.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.1.3.1.2.1

Move $x$ .

$x^{2} + 7 x + 6 = 2 \cdot 2 (x \cdot x) + 2 x \sqrt{(x + 1) (x + 6)} + \sqrt{(x + 1) (x + 6)} (2 x) + \sqrt{(x + 1) (x + 6)} \sqrt{(x + 1) (x + 6)}$

Step 5.3.1.3.1.2.2

Multiply $x$ by $x$ .

$x^{2} + 7 x + 6 = 2 \cdot 2 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + \sqrt{(x + 1) (x + 6)} (2 x) + \sqrt{(x + 1) (x + 6)} \sqrt{(x + 1) (x + 6)}$

Step 5.3.1.3.1.3

Multiply $2$ by $2$ .

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + \sqrt{(x + 1) (x + 6)} (2 x) + \sqrt{(x + 1) (x + 6)} \sqrt{(x + 1) (x + 6)}$

Step 5.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + \sqrt{(x + 1) (x + 6)} \sqrt{(x + 1) (x + 6)}$

Step 5.3.1.3.1.5

Multiply $\sqrt{(x + 1) (x + 6)} \sqrt{(x + 1) (x + 6)}$ .

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Step 5.3.1.3.1.5.1

Raise $\sqrt{(x + 1) (x + 6)}$ to the power of $1$ .

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + {\sqrt{(x + 1) (x + 6)}}^{1} \sqrt{(x + 1) (x + 6)}$

Step 5.3.1.3.1.5.2

Raise $\sqrt{(x + 1) (x + 6)}$ to the power of $1$ .

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + {\sqrt{(x + 1) (x + 6)}}^{1} {\sqrt{(x + 1) (x + 6)}}^{1}$

Step 5.3.1.3.1.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + {\sqrt{(x + 1) (x + 6)}}^{1 + 1}$

Step 5.3.1.3.1.5.4

Add $1$ and $1$ .

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + {\sqrt{(x + 1) (x + 6)}}^{2}$

Step 5.3.1.3.1.6

Rewrite ${\sqrt{(x + 1) (x + 6)}}^{2}$ as $(x + 1) (x + 6)$ .

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Step 5.3.1.3.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x + 1) (x + 6)}$ as ${((x + 1) (x + 6))}^{\frac{1}{2}}$ .

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + {({((x + 1) (x + 6))}^{\frac{1}{2}})}^{2}$

Step 5.3.1.3.1.6.2

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

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + {((x + 1) (x + 6))}^{\frac{1}{2} \cdot 2}$

Step 5.3.1.3.1.6.3

Combine $\frac{1}{2}$ and $2$ .

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + {((x + 1) (x + 6))}^{\frac{2}{2}}$

Step 5.3.1.3.1.6.4

Cancel the common factor of $2$ .

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Step 5.3.1.3.1.6.4.1

Cancel the common factor.

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + {((x + 1) (x + 6))}^{\frac{2}{2}}$

Step 5.3.1.3.1.6.4.2

Rewrite the expression.

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + {((x + 1) (x + 6))}^{1}$

Step 5.3.1.3.1.6.5

Simplify.

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + (x + 1) (x + 6)$

Step 5.3.1.3.1.7

Expand $(x + 1) (x + 6)$ using the FOIL Method.

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Step 5.3.1.3.1.7.1

Apply the distributive property.

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + x (x + 6) + 1 (x + 6)$

Step 5.3.1.3.1.7.2

Apply the distributive property.

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + x \cdot x + x \cdot 6 + 1 (x + 6)$

Step 5.3.1.3.1.7.3

Apply the distributive property.

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + x \cdot x + x \cdot 6 + 1 x + 1 \cdot 6$

Step 5.3.1.3.1.8

Simplify and combine like terms.

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Step 5.3.1.3.1.8.1

Simplify each term.

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Step 5.3.1.3.1.8.1.1

Multiply $x$ by $x$ .

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + x^{2} + x \cdot 6 + 1 x + 1 \cdot 6$

Step 5.3.1.3.1.8.1.2

Move $6$ to the left of $x$ .

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + x^{2} + 6 \cdot x + 1 x + 1 \cdot 6$

Step 5.3.1.3.1.8.1.3

Multiply $x$ by $1$ .

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + x^{2} + 6 x + x + 1 \cdot 6$

Step 5.3.1.3.1.8.1.4

Multiply $6$ by $1$ .

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + x^{2} + 6 x + x + 6$

Step 5.3.1.3.1.8.2

Add $6 x$ and $x$ .

$x^{2} + 7 x + 6 = 4 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + x^{2} + 7 x + 6$

Step 5.3.1.3.2

Add $4 x^{2}$ and $x^{2}$ .

$x^{2} + 7 x + 6 = 5 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 \sqrt{(x + 1) (x + 6)} x + 7 x + 6$

Step 5.3.1.3.3

Reorder the factors of $2 \sqrt{(x + 1) (x + 6)} x$ .

$x^{2} + 7 x + 6 = 5 x^{2} + 2 x \sqrt{(x + 1) (x + 6)} + 2 x \sqrt{(x + 1) (x + 6)} + 7 x + 6$

Step 5.3.1.3.4

Add $2 x \sqrt{(x + 1) (x + 6)}$ and $2 x \sqrt{(x + 1) (x + 6)}$ .

$x^{2} + 7 x + 6 = 5 x^{2} + 4 x \sqrt{(x + 1) (x + 6)} + 7 x + 6$

Step 6

Solve for $4 x \sqrt{(x + 1) (x + 6)}$ .

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Step 6.1

Rewrite the equation as $5 x^{2} + 4 x \sqrt{(x + 1) (x + 6)} + 7 x + 6 = x^{2} + 7 x + 6$ .

$5 x^{2} + 4 x \sqrt{(x + 1) (x + 6)} + 7 x + 6 = x^{2} + 7 x + 6$

Step 6.2

Move all terms not containing $4 x \sqrt{(x + 1) (x + 6)}$ to the right side of the equation.

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Step 6.2.1

Subtract $5 x^{2}$ from both sides of the equation.

$4 x \sqrt{(x + 1) (x + 6)} + 7 x + 6 = x^{2} + 7 x + 6 - 5 x^{2}$

Step 6.2.2

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

$4 x \sqrt{(x + 1) (x + 6)} + 6 = x^{2} + 7 x + 6 - 5 x^{2} - 7 x$

Step 6.2.3

Subtract $6$ from both sides of the equation.

$4 x \sqrt{(x + 1) (x + 6)} = x^{2} + 7 x + 6 - 5 x^{2} - 7 x - 6$

Step 6.2.4

Combine the opposite terms in $x^{2} + 7 x + 6 - 5 x^{2} - 7 x - 6$ .

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Step 6.2.4.1

Subtract $7 x$ from $7 x$ .

$4 x \sqrt{(x + 1) (x + 6)} = x^{2} + 6 - 5 x^{2} + 0 - 6$

Step 6.2.4.2

Add $x^{2} + 6 - 5 x^{2}$ and $0$ .

$4 x \sqrt{(x + 1) (x + 6)} = x^{2} + 6 - 5 x^{2} - 6$

Step 6.2.4.3

Subtract $6$ from $6$ .

$4 x \sqrt{(x + 1) (x + 6)} = x^{2} - 5 x^{2} + 0$

Step 6.2.4.4

Add $x^{2} - 5 x^{2}$ and $0$ .

$4 x \sqrt{(x + 1) (x + 6)} = x^{2} - 5 x^{2}$

Step 6.2.5

Subtract $5 x^{2}$ from $x^{2}$ .

$4 x \sqrt{(x + 1) (x + 6)} = - 4 x^{2}$

Step 7

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

${(4 x \sqrt{(x + 1) (x + 6)})}^{2} = {(- 4 x^{2})}^{2}$

Step 8

Simplify each side of the equation.

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Step 8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x + 1) (x + 6)}$ as ${((x + 1) (x + 6))}^{\frac{1}{2}}$ .

${(4 x {((x + 1) (x + 6))}^{\frac{1}{2}})}^{2} = {(- 4 x^{2})}^{2}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Simplify ${(4 x {((x + 1) (x + 6))}^{\frac{1}{2}})}^{2}$ .

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Step 8.2.1.1

Expand $(x + 1) (x + 6)$ using the FOIL Method.

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Step 8.2.1.1.1

Apply the distributive property.

${(4 x {(x (x + 6) + 1 (x + 6))}^{\frac{1}{2}})}^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.1.2

Apply the distributive property.

${(4 x {(x \cdot x + x \cdot 6 + 1 (x + 6))}^{\frac{1}{2}})}^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.1.3

Apply the distributive property.

${(4 x {(x \cdot x + x \cdot 6 + 1 x + 1 \cdot 6)}^{\frac{1}{2}})}^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.2

Simplify and combine like terms.

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Step 8.2.1.2.1

Simplify each term.

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Step 8.2.1.2.1.1

Multiply $x$ by $x$ .

${(4 x {(x^{2} + x \cdot 6 + 1 x + 1 \cdot 6)}^{\frac{1}{2}})}^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.2.1.2

Move $6$ to the left of $x$ .

${(4 x {(x^{2} + 6 \cdot x + 1 x + 1 \cdot 6)}^{\frac{1}{2}})}^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.2.1.3

Multiply $x$ by $1$ .

${(4 x {(x^{2} + 6 x + x + 1 \cdot 6)}^{\frac{1}{2}})}^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.2.1.4

Multiply $6$ by $1$ .

${(4 x {(x^{2} + 6 x + x + 6)}^{\frac{1}{2}})}^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.2.2

Add $6 x$ and $x$ .

${(4 x {(x^{2} + 7 x + 6)}^{\frac{1}{2}})}^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.2.1.3.1

Apply the product rule to $4 x {(x^{2} + 7 x + 6)}^{\frac{1}{2}}$ .

${(4 x)}^{2} {({(x^{2} + 7 x + 6)}^{\frac{1}{2}})}^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.3.2

Apply the product rule to $4 x$ .

$4^{2} x^{2} {({(x^{2} + 7 x + 6)}^{\frac{1}{2}})}^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.4

Raise $4$ to the power of $2$ .

$16 x^{2} {({(x^{2} + 7 x + 6)}^{\frac{1}{2}})}^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.5

Multiply the exponents in ${({(x^{2} + 7 x + 6)}^{\frac{1}{2}})}^{2}$ .

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Step 8.2.1.5.1

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

$16 x^{2} {(x^{2} + 7 x + 6)}^{\frac{1}{2} \cdot 2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.5.2

Cancel the common factor of $2$ .

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Step 8.2.1.5.2.1

Cancel the common factor.

$16 x^{2} {(x^{2} + 7 x + 6)}^{\frac{1}{2} \cdot 2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.5.2.2

Rewrite the expression.

$16 x^{2} {(x^{2} + 7 x + 6)}^{1} = {(- 4 x^{2})}^{2}$

Step 8.2.1.6

Simplify.

$16 x^{2} (x^{2} + 7 x + 6) = {(- 4 x^{2})}^{2}$

Step 8.2.1.7

Apply the distributive property.

$16 x^{2} x^{2} + 16 x^{2} (7 x) + 16 x^{2} \cdot 6 = {(- 4 x^{2})}^{2}$

Step 8.2.1.8

Simplify.

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Step 8.2.1.8.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 8.2.1.8.1.1

Move $x^{2}$ .

$16 (x^{2} x^{2}) + 16 x^{2} (7 x) + 16 x^{2} \cdot 6 = {(- 4 x^{2})}^{2}$

Step 8.2.1.8.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$16 x^{2 + 2} + 16 x^{2} (7 x) + 16 x^{2} \cdot 6 = {(- 4 x^{2})}^{2}$

Step 8.2.1.8.1.3

Add $2$ and $2$ .

$16 x^{4} + 16 x^{2} (7 x) + 16 x^{2} \cdot 6 = {(- 4 x^{2})}^{2}$

Step 8.2.1.8.2

Rewrite using the commutative property of multiplication.

$16 x^{4} + 16 \cdot 7 x^{2} x + 16 x^{2} \cdot 6 = {(- 4 x^{2})}^{2}$

Step 8.2.1.8.3

Multiply $6$ by $16$ .

$16 x^{4} + 16 \cdot 7 x^{2} x + 96 x^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.9

Simplify each term.

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Step 8.2.1.9.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 8.2.1.9.1.1

Move $x$ .

$16 x^{4} + 16 \cdot 7 (x \cdot x^{2}) + 96 x^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.9.1.2

Multiply $x$ by $x^{2}$ .

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Step 8.2.1.9.1.2.1

Raise $x$ to the power of $1$ .

$16 x^{4} + 16 \cdot 7 (x^{1} x^{2}) + 96 x^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.9.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$16 x^{4} + 16 \cdot 7 x^{1 + 2} + 96 x^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.9.1.3

Add $1$ and $2$ .

$16 x^{4} + 16 \cdot 7 x^{3} + 96 x^{2} = {(- 4 x^{2})}^{2}$

Step 8.2.1.9.2

Multiply $16$ by $7$ .

$16 x^{4} + 112 x^{3} + 96 x^{2} = {(- 4 x^{2})}^{2}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify ${(- 4 x^{2})}^{2}$ .

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Step 8.3.1.1

Apply the product rule to $- 4 x^{2}$ .

$16 x^{4} + 112 x^{3} + 96 x^{2} = {(- 4)}^{2} {(x^{2})}^{2}$

Step 8.3.1.2

Raise $- 4$ to the power of $2$ .

$16 x^{4} + 112 x^{3} + 96 x^{2} = 16 {(x^{2})}^{2}$

Step 8.3.1.3

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 8.3.1.3.1

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

$16 x^{4} + 112 x^{3} + 96 x^{2} = 16 x^{2 \cdot 2}$

Step 8.3.1.3.2

Multiply $2$ by $2$ .

$16 x^{4} + 112 x^{3} + 96 x^{2} = 16 x^{4}$

Step 9

Solve for $x$ .

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Step 9.1

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

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Step 9.1.1

Subtract $16 x^{4}$ from both sides of the equation.

$16 x^{4} + 112 x^{3} + 96 x^{2} - 16 x^{4} = 0$

Step 9.1.2

Combine the opposite terms in $16 x^{4} + 112 x^{3} + 96 x^{2} - 16 x^{4}$ .

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Step 9.1.2.1

Subtract $16 x^{4}$ from $16 x^{4}$ .

$112 x^{3} + 96 x^{2} + 0 = 0$

Step 9.1.2.2

Add $112 x^{3} + 96 x^{2}$ and $0$ .

$112 x^{3} + 96 x^{2} = 0$

Step 9.2

Factor $16 x^{2}$ out of $112 x^{3} + 96 x^{2}$ .

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Step 9.2.1

Factor $16 x^{2}$ out of $112 x^{3}$ .

$16 x^{2} (7 x) + 96 x^{2} = 0$

Step 9.2.2

Factor $16 x^{2}$ out of $96 x^{2}$ .

$16 x^{2} (7 x) + 16 x^{2} (6) = 0$

Step 9.2.3

Factor $16 x^{2}$ out of $16 x^{2} (7 x) + 16 x^{2} (6)$ .

$16 x^{2} (7 x + 6) = 0$

Step 9.3

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

$x^{2} = 0$

$7 x + 6 = 0$

Step 9.4

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 9.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 9.4.2

Solve $x^{2} = 0$ for $x$ .

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Step 9.4.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 9.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 9.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 9.4.2.2.2

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

$x = \pm 0$

Step 9.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 9.5

Set $7 x + 6$ equal to $0$ and solve for $x$ .

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Step 9.5.1

Set $7 x + 6$ equal to $0$ .

$7 x + 6 = 0$

Step 9.5.2

Solve $7 x + 6 = 0$ for $x$ .

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Step 9.5.2.1

Subtract $6$ from both sides of the equation.

$7 x = - 6$

Step 9.5.2.2

Divide each term in $7 x = - 6$ by $7$ and simplify.

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Step 9.5.2.2.1

Divide each term in $7 x = - 6$ by $7$ .

$\frac{7 x}{7} = \frac{- 6}{7}$

Step 9.5.2.2.2

Simplify the left side.

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Step 9.5.2.2.2.1

Cancel the common factor of $7$ .

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Step 9.5.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 6}{7}$

Step 9.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{7}$

Step 9.5.2.2.3

Simplify the right side.

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Step 9.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{6}{7}$

Step 9.6

The final solution is all the values that make $16 x^{2} (7 x + 6) = 0$ true.

$x = 0, - \frac{6}{7}$

Step 10

Exclude the solutions that do not make $\sqrt{4 x + 7} = \sqrt{x + 1} - \sqrt{x + 6}$ true.

$No solution$