Algebra Examples

$\sqrt{7 x + 1} - \sqrt{x - 5} = 6$

Step 1

Add $\sqrt{x - 5}$ to both sides of the equation.

$\sqrt{7 x + 1} = 6 + \sqrt{x - 5}$

Step 2

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

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

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x + 1}$ as ${(7 x + 1)}^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(6 + \sqrt{x - 5})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(6 + \sqrt{x - 5})}^{2}$ as $(6 + \sqrt{x - 5}) (6 + \sqrt{x - 5})$ .

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

Step 3.3.1.2

Expand $(6 + \sqrt{x - 5}) (6 + \sqrt{x - 5})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

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

Step 3.3.1.2.2

Apply the distributive property.

$7 x + 1 = 6 \cdot 6 + 6 \sqrt{x - 5} + \sqrt{x - 5} (6 + \sqrt{x - 5})$

Step 3.3.1.2.3

Apply the distributive property.

$7 x + 1 = 6 \cdot 6 + 6 \sqrt{x - 5} + \sqrt{x - 5} \cdot 6 + \sqrt{x - 5} \sqrt{x - 5}$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $6$ by $6$ .

$7 x + 1 = 36 + 6 \sqrt{x - 5} + \sqrt{x - 5} \cdot 6 + \sqrt{x - 5} \sqrt{x - 5}$

Step 3.3.1.3.1.2

Move $6$ to the left of $\sqrt{x - 5}$ .

$7 x + 1 = 36 + 6 \sqrt{x - 5} + 6 \cdot \sqrt{x - 5} + \sqrt{x - 5} \sqrt{x - 5}$

Step 3.3.1.3.1.3

Multiply $\sqrt{x - 5} \sqrt{x - 5}$ .

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Step 3.3.1.3.1.3.1

Raise $\sqrt{x - 5}$ to the power of $1$ .

$7 x + 1 = 36 + 6 \sqrt{x - 5} + 6 \sqrt{x - 5} + {\sqrt{x - 5}}^{1} \sqrt{x - 5}$

Step 3.3.1.3.1.3.2

Raise $\sqrt{x - 5}$ to the power of $1$ .

$7 x + 1 = 36 + 6 \sqrt{x - 5} + 6 \sqrt{x - 5} + {\sqrt{x - 5}}^{1} {\sqrt{x - 5}}^{1}$

Step 3.3.1.3.1.3.3

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

$7 x + 1 = 36 + 6 \sqrt{x - 5} + 6 \sqrt{x - 5} + {\sqrt{x - 5}}^{1 + 1}$

Step 3.3.1.3.1.3.4

Add $1$ and $1$ .

$7 x + 1 = 36 + 6 \sqrt{x - 5} + 6 \sqrt{x - 5} + {\sqrt{x - 5}}^{2}$

Step 3.3.1.3.1.4

Rewrite ${\sqrt{x - 5}}^{2}$ as $x - 5$ .

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Step 3.3.1.3.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 5}$ as ${(x - 5)}^{\frac{1}{2}}$ .

$7 x + 1 = 36 + 6 \sqrt{x - 5} + 6 \sqrt{x - 5} + {({(x - 5)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.4.2

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

$7 x + 1 = 36 + 6 \sqrt{x - 5} + 6 \sqrt{x - 5} + {(x - 5)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.4.3

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

$7 x + 1 = 36 + 6 \sqrt{x - 5} + 6 \sqrt{x - 5} + {(x - 5)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.4.4.1

Cancel the common factor.

$7 x + 1 = 36 + 6 \sqrt{x - 5} + 6 \sqrt{x - 5} + {(x - 5)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4.2

Rewrite the expression.

$7 x + 1 = 36 + 6 \sqrt{x - 5} + 6 \sqrt{x - 5} + {(x - 5)}^{1}$

Step 3.3.1.3.1.4.5

Simplify.

$7 x + 1 = 36 + 6 \sqrt{x - 5} + 6 \sqrt{x - 5} + x - 5$

Step 3.3.1.3.2

Subtract $5$ from $36$ .

$7 x + 1 = 31 + 6 \sqrt{x - 5} + 6 \sqrt{x - 5} + x$

Step 3.3.1.3.3

Add $6 \sqrt{x - 5}$ and $6 \sqrt{x - 5}$ .

$7 x + 1 = 31 + 12 \sqrt{x - 5} + x$

Step 4

Solve for $12 \sqrt{x - 5}$ .

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Step 4.1

Rewrite the equation as $31 + 12 \sqrt{x - 5} + x = 7 x + 1$ .

$31 + 12 \sqrt{x - 5} + x = 7 x + 1$

Step 4.2

Move all terms not containing $12 \sqrt{x - 5}$ to the right side of the equation.

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Step 4.2.1

Subtract $31$ from both sides of the equation.

$12 \sqrt{x - 5} + x = 7 x + 1 - 31$

Step 4.2.2

Subtract $x$ from both sides of the equation.

$12 \sqrt{x - 5} = 7 x + 1 - 31 - x$

Step 4.2.3

Subtract $x$ from $7 x$ .

$12 \sqrt{x - 5} = 6 x + 1 - 31$

Step 4.2.4

Subtract $31$ from $1$ .

$12 \sqrt{x - 5} = 6 x - 30$

Step 5

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

${(12 \sqrt{x - 5})}^{2} = {(6 x - 30)}^{2}$

Step 6

Simplify each side of the equation.

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 5}$ as ${(x - 5)}^{\frac{1}{2}}$ .

${(12 {(x - 5)}^{\frac{1}{2}})}^{2} = {(6 x - 30)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(12 {(x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $12 {(x - 5)}^{\frac{1}{2}}$ .

$12^{2} {({(x - 5)}^{\frac{1}{2}})}^{2} = {(6 x - 30)}^{2}$

Step 6.2.1.2

Raise $12$ to the power of $2$ .

$144 {({(x - 5)}^{\frac{1}{2}})}^{2} = {(6 x - 30)}^{2}$

Step 6.2.1.3

Multiply the exponents in ${({(x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.3.1

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

$144 {(x - 5)}^{\frac{1}{2} \cdot 2} = {(6 x - 30)}^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

$144 {(x - 5)}^{\frac{1}{2} \cdot 2} = {(6 x - 30)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$144 {(x - 5)}^{1} = {(6 x - 30)}^{2}$

Step 6.2.1.4

Simplify.

$144 (x - 5) = {(6 x - 30)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$144 x + 144 \cdot - 5 = {(6 x - 30)}^{2}$

Step 6.2.1.6

Multiply $144$ by $- 5$ .

$144 x - 720 = {(6 x - 30)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(6 x - 30)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(6 x - 30)}^{2}$ as $(6 x - 30) (6 x - 30)$ .

$144 x - 720 = (6 x - 30) (6 x - 30)$

Step 6.3.1.2

Expand $(6 x - 30) (6 x - 30)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$144 x - 720 = 6 x (6 x - 30) - 30 (6 x - 30)$

Step 6.3.1.2.2

Apply the distributive property.

$144 x - 720 = 6 x (6 x) + 6 x \cdot - 30 - 30 (6 x - 30)$

Step 6.3.1.2.3

Apply the distributive property.

$144 x - 720 = 6 x (6 x) + 6 x \cdot - 30 - 30 (6 x) - 30 \cdot - 30$

Step 6.3.1.3

Simplify and combine like terms.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$144 x - 720 = 6 \cdot 6 x \cdot x + 6 x \cdot - 30 - 30 (6 x) - 30 \cdot - 30$

Step 6.3.1.3.1.2

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

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Step 6.3.1.3.1.2.1

Move $x$ .

$144 x - 720 = 6 \cdot 6 (x \cdot x) + 6 x \cdot - 30 - 30 (6 x) - 30 \cdot - 30$

Step 6.3.1.3.1.2.2

Multiply $x$ by $x$ .

$144 x - 720 = 6 \cdot 6 x^{2} + 6 x \cdot - 30 - 30 (6 x) - 30 \cdot - 30$

Step 6.3.1.3.1.3

Multiply $6$ by $6$ .

$144 x - 720 = 36 x^{2} + 6 x \cdot - 30 - 30 (6 x) - 30 \cdot - 30$

Step 6.3.1.3.1.4

Multiply $- 30$ by $6$ .

$144 x - 720 = 36 x^{2} - 180 x - 30 (6 x) - 30 \cdot - 30$

Step 6.3.1.3.1.5

Multiply $6$ by $- 30$ .

$144 x - 720 = 36 x^{2} - 180 x - 180 x - 30 \cdot - 30$

Step 6.3.1.3.1.6

Multiply $- 30$ by $- 30$ .

$144 x - 720 = 36 x^{2} - 180 x - 180 x + 900$

Step 6.3.1.3.2

Subtract $180 x$ from $- 180 x$ .

$144 x - 720 = 36 x^{2} - 360 x + 900$

Step 7

Solve for $x$ .

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Step 7.1

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

$36 x^{2} - 360 x + 900 = 144 x - 720$

Step 7.2

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

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Step 7.2.1

Subtract $144 x$ from both sides of the equation.

$36 x^{2} - 360 x + 900 - 144 x = - 720$

Step 7.2.2

Subtract $144 x$ from $- 360 x$ .

$36 x^{2} - 504 x + 900 = - 720$

Step 7.3

Add $720$ to both sides of the equation.

$36 x^{2} - 504 x + 900 + 720 = 0$

Step 7.4

Add $900$ and $720$ .

$36 x^{2} - 504 x + 1620 = 0$

Step 7.5

Factor the left side of the equation.

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Step 7.5.1

Factor $36$ out of $36 x^{2} - 504 x + 1620$ .

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Step 7.5.1.1

Factor $36$ out of $36 x^{2}$ .

$36 (x^{2}) - 504 x + 1620 = 0$

Step 7.5.1.2

Factor $36$ out of $- 504 x$ .

$36 (x^{2}) + 36 (- 14 x) + 1620 = 0$

Step 7.5.1.3

Factor $36$ out of $1620$ .

$36 x^{2} + 36 (- 14 x) + 36 \cdot 45 = 0$

Step 7.5.1.4

Factor $36$ out of $36 x^{2} + 36 (- 14 x)$ .

$36 (x^{2} - 14 x) + 36 \cdot 45 = 0$

Step 7.5.1.5

Factor $36$ out of $36 (x^{2} - 14 x) + 36 \cdot 45$ .

$36 (x^{2} - 14 x + 45) = 0$

Step 7.5.2

Factor.

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Step 7.5.2.1

Factor $x^{2} - 14 x + 45$ using the AC method.

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Step 7.5.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $45$ and whose sum is $- 14$ .

$- 9, - 5$

Step 7.5.2.1.2

Write the factored form using these integers.

$36 ((x - 9) (x - 5)) = 0$

Step 7.5.2.2

Remove unnecessary parentheses.

$36 (x - 9) (x - 5) = 0$

Step 7.6

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

$x - 9 = 0$

$x - 5 = 0$

Step 7.7

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 7.7.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 7.7.2

Add $9$ to both sides of the equation.

$x = 9$

Step 7.8

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 7.8.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 7.8.2

Add $5$ to both sides of the equation.

$x = 5$

Step 7.9

The final solution is all the values that make $36 (x - 9) (x - 5) = 0$ true.

$x = 9, 5$