Algebra Examples

$(\sqrt{x - 6} - 11) (4 - 5 \sqrt{x - 1}) = 0$

Step 1

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

$\sqrt{x - 6} - 11 = 0$

$4 - 5 \sqrt{x - 1} = 0$

Step 2

Set $\sqrt{x - 6} - 11$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $\sqrt{x - 6} - 11$ equal to $0$ .

$\sqrt{x - 6} - 11 = 0$

Step 2.2

Solve $\sqrt{x - 6} - 11 = 0$ for $x$ .

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Step 2.2.1

Add $11$ to both sides of the equation.

$\sqrt{x - 6} = 11$

Step 2.2.2

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

${\sqrt{x - 6}}^{2} = 11^{2}$

Step 2.2.3

Simplify each side of the equation.

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Step 2.2.3.1

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

${({(x - 6)}^{\frac{1}{2}})}^{2} = 11^{2}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

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

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Step 2.2.3.2.1.1

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

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Step 2.2.3.2.1.1.1

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

${(x - 6)}^{\frac{1}{2} \cdot 2} = 11^{2}$

Step 2.2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.3.2.1.1.2.1

Cancel the common factor.

${(x - 6)}^{\frac{1}{2} \cdot 2} = 11^{2}$

Step 2.2.3.2.1.1.2.2

Rewrite the expression.

${(x - 6)}^{1} = 11^{2}$

Step 2.2.3.2.1.2

Simplify.

$x - 6 = 11^{2}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Raise $11$ to the power of $2$ .

$x - 6 = 121$

Step 2.2.4

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

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Step 2.2.4.1

Add $6$ to both sides of the equation.

$x = 121 + 6$

Step 2.2.4.2

Add $121$ and $6$ .

$x = 127$

Step 3

Set $4 - 5 \sqrt{x - 1}$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $4 - 5 \sqrt{x - 1}$ equal to $0$ .

$4 - 5 \sqrt{x - 1} = 0$

Step 3.2

Solve $4 - 5 \sqrt{x - 1} = 0$ for $x$ .

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Step 3.2.1

Subtract $4$ from both sides of the equation.

$- 5 \sqrt{x - 1} = - 4$

Step 3.2.2

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

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

Step 3.2.3

Simplify each side of the equation.

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Step 3.2.3.1

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

${(- 5 {(x - 1)}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

Step 3.2.3.2

Simplify the left side.

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Step 3.2.3.2.1

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

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Step 3.2.3.2.1.1

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

${(- 5)}^{2} {({(x - 1)}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

Step 3.2.3.2.1.2

Raise $- 5$ to the power of $2$ .

$25 {({(x - 1)}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

Step 3.2.3.2.1.3

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

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Step 3.2.3.2.1.3.1

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

$25 {(x - 1)}^{\frac{1}{2} \cdot 2} = {(- 4)}^{2}$

Step 3.2.3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.3.2.1.3.2.1

Cancel the common factor.

$25 {(x - 1)}^{\frac{1}{2} \cdot 2} = {(- 4)}^{2}$

Step 3.2.3.2.1.3.2.2

Rewrite the expression.

$25 {(x - 1)}^{1} = {(- 4)}^{2}$

Step 3.2.3.2.1.4

Simplify.

$25 (x - 1) = {(- 4)}^{2}$

Step 3.2.3.2.1.5

Apply the distributive property.

$25 x + 25 \cdot - 1 = {(- 4)}^{2}$

Step 3.2.3.2.1.6

Multiply $25$ by $- 1$ .

$25 x - 25 = {(- 4)}^{2}$

Step 3.2.3.3

Simplify the right side.

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Step 3.2.3.3.1

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

$25 x - 25 = 16$

Step 3.2.4

Solve for $x$ .

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Step 3.2.4.1

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

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Step 3.2.4.1.1

Add $25$ to both sides of the equation.

$25 x = 16 + 25$

Step 3.2.4.1.2

Add $16$ and $25$ .

$25 x = 41$

Step 3.2.4.2

Divide each term in $25 x = 41$ by $25$ and simplify.

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Step 3.2.4.2.1

Divide each term in $25 x = 41$ by $25$ .

$\frac{25 x}{25} = \frac{41}{25}$

Step 3.2.4.2.2

Simplify the left side.

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Step 3.2.4.2.2.1

Cancel the common factor of $25$ .

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Step 3.2.4.2.2.1.1

Cancel the common factor.

$\frac{25 x}{25} = \frac{41}{25}$

Step 3.2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{41}{25}$

Step 4

The final solution is all the values that make $(\sqrt{x - 6} - 11) (4 - 5 \sqrt{x - 1}) = 0$ true.

$x = 127, \frac{41}{25}$

Step 5

Exclude the solutions that do not make $(\sqrt{x - 6} - 11) (4 - 5 \sqrt{x - 1}) = 0$ true.

$x = 127$