Algebra Examples

${(5 - \sqrt{x})}^{2} = y - 20 \sqrt{2}$

Step 1

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

$5 - \sqrt{x} = \pm \sqrt{y - 20 \sqrt{2}}$

Step 2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.1

First, use the positive value of the $\pm$ to find the first solution.

$5 - \sqrt{x} = \sqrt{y - 20 \sqrt{2}}$

Step 2.2

Subtract $5$ from both sides of the equation.

$- \sqrt{x} = \sqrt{y - 20 \sqrt{2}} - 5$

Step 2.3

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

${(- \sqrt{x})}^{2} = {(\sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.4

Simplify each side of the equation.

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Step 2.4.1

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

${(- x^{\frac{1}{2}})}^{2} = {(\sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

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

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Step 2.4.2.1.1

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

${(- 1)}^{2} {(x^{\frac{1}{2}})}^{2} = {(\sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.4.2.1.2

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

$1 {(x^{\frac{1}{2}})}^{2} = {(\sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.4.2.1.3

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

${(x^{\frac{1}{2}})}^{2} = {(\sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.4.2.1.4

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

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Step 2.4.2.1.4.1

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

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.4.2.1.4.2

Cancel the common factor of $2$ .

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Step 2.4.2.1.4.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.4.2.1.4.2.2

Rewrite the expression.

$x^{1} = {(\sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.4.2.1.5

Simplify.

$x = {(\sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify ${(\sqrt{y - 20 \sqrt{2}} - 5)}^{2}$ .

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Step 2.4.3.1.1

Rewrite ${(\sqrt{y - 20 \sqrt{2}} - 5)}^{2}$ as $(\sqrt{y - 20 \sqrt{2}} - 5) (\sqrt{y - 20 \sqrt{2}} - 5)$ .

$x = (\sqrt{y - 20 \sqrt{2}} - 5) (\sqrt{y - 20 \sqrt{2}} - 5)$

Step 2.4.3.1.2

Expand $(\sqrt{y - 20 \sqrt{2}} - 5) (\sqrt{y - 20 \sqrt{2}} - 5)$ using the FOIL Method.

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Step 2.4.3.1.2.1

Apply the distributive property.

$x = \sqrt{y - 20 \sqrt{2}} (\sqrt{y - 20 \sqrt{2}} - 5) - 5 (\sqrt{y - 20 \sqrt{2}} - 5)$

Step 2.4.3.1.2.2

Apply the distributive property.

$x = \sqrt{y - 20 \sqrt{2}} \sqrt{y - 20 \sqrt{2}} + \sqrt{y - 20 \sqrt{2}} \cdot - 5 - 5 (\sqrt{y - 20 \sqrt{2}} - 5)$

Step 2.4.3.1.2.3

Apply the distributive property.

$x = \sqrt{y - 20 \sqrt{2}} \sqrt{y - 20 \sqrt{2}} + \sqrt{y - 20 \sqrt{2}} \cdot - 5 - 5 \sqrt{y - 20 \sqrt{2}} - 5 \cdot - 5$

Step 2.4.3.1.3

Simplify and combine like terms.

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Step 2.4.3.1.3.1

Simplify each term.

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Step 2.4.3.1.3.1.1

Multiply $\sqrt{y - 20 \sqrt{2}}$ by $\sqrt{y - 20 \sqrt{2}}$ .

$x = {\sqrt{y - 20 \sqrt{2}}}^{2} + \sqrt{y - 20 \sqrt{2}} \cdot - 5 - 5 \sqrt{y - 20 \sqrt{2}} - 5 \cdot - 5$

Step 2.4.3.1.3.1.2

Move $- 5$ to the left of $\sqrt{y - 20 \sqrt{2}}$ .

$x = {\sqrt{y - 20 \sqrt{2}}}^{2} - 5 \cdot \sqrt{y - 20 \sqrt{2}} - 5 \sqrt{y - 20 \sqrt{2}} - 5 \cdot - 5$

Step 2.4.3.1.3.1.3

Multiply $- 5$ by $- 5$ .

$x = {\sqrt{y - 20 \sqrt{2}}}^{2} - 5 \sqrt{y - 20 \sqrt{2}} - 5 \sqrt{y - 20 \sqrt{2}} + 25$

Step 2.4.3.1.3.2

Subtract $5 \sqrt{y - 20 \sqrt{2}}$ from $- 5 \sqrt{y - 20 \sqrt{2}}$ .

$x = {\sqrt{y - 20 \sqrt{2}}}^{2} - 10 \sqrt{y - 20 \sqrt{2}} + 25$

Step 2.5

Next, use the negative value of the $\pm$ to find the second solution.

$5 - \sqrt{x} = - \sqrt{y - 20 \sqrt{2}}$

Step 2.6

Subtract $5$ from both sides of the equation.

$- \sqrt{x} = - \sqrt{y - 20 \sqrt{2}} - 5$

Step 2.7

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

${(- \sqrt{x})}^{2} = {(- \sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.8

Simplify each side of the equation.

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Step 2.8.1

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

${(- x^{\frac{1}{2}})}^{2} = {(- \sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.8.2

Simplify the left side.

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Step 2.8.2.1

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

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Step 2.8.2.1.1

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

${(- 1)}^{2} {(x^{\frac{1}{2}})}^{2} = {(- \sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.8.2.1.2

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

$1 {(x^{\frac{1}{2}})}^{2} = {(- \sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.8.2.1.3

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

${(x^{\frac{1}{2}})}^{2} = {(- \sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.8.2.1.4

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

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Step 2.8.2.1.4.1

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

$x^{\frac{1}{2} \cdot 2} = {(- \sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.8.2.1.4.2

Cancel the common factor of $2$ .

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Step 2.8.2.1.4.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(- \sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.8.2.1.4.2.2

Rewrite the expression.

$x^{1} = {(- \sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.8.2.1.5

Simplify.

$x = {(- \sqrt{y - 20 \sqrt{2}} - 5)}^{2}$

Step 2.8.3

Simplify the right side.

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Step 2.8.3.1

Simplify ${(- \sqrt{y - 20 \sqrt{2}} - 5)}^{2}$ .

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Step 2.8.3.1.1

Rewrite ${(- \sqrt{y - 20 \sqrt{2}} - 5)}^{2}$ as $(- \sqrt{y - 20 \sqrt{2}} - 5) (- \sqrt{y - 20 \sqrt{2}} - 5)$ .

$x = (- \sqrt{y - 20 \sqrt{2}} - 5) (- \sqrt{y - 20 \sqrt{2}} - 5)$

Step 2.8.3.1.2

Expand $(- \sqrt{y - 20 \sqrt{2}} - 5) (- \sqrt{y - 20 \sqrt{2}} - 5)$ using the FOIL Method.

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Step 2.8.3.1.2.1

Apply the distributive property.

$x = - \sqrt{y - 20 \sqrt{2}} (- \sqrt{y - 20 \sqrt{2}} - 5) - 5 (- \sqrt{y - 20 \sqrt{2}} - 5)$

Step 2.8.3.1.2.2

Apply the distributive property.

$x = - \sqrt{y - 20 \sqrt{2}} (- \sqrt{y - 20 \sqrt{2}}) - \sqrt{y - 20 \sqrt{2}} \cdot - 5 - 5 (- \sqrt{y - 20 \sqrt{2}} - 5)$

Step 2.8.3.1.2.3

Apply the distributive property.

$x = - \sqrt{y - 20 \sqrt{2}} (- \sqrt{y - 20 \sqrt{2}}) - \sqrt{y - 20 \sqrt{2}} \cdot - 5 - 5 (- \sqrt{y - 20 \sqrt{2}}) - 5 \cdot - 5$

Step 2.8.3.1.3

Simplify and combine like terms.

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Step 2.8.3.1.3.1

Simplify each term.

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Step 2.8.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$x = - 1 \cdot - 1 \sqrt{y - 20 \sqrt{2}} \sqrt{y - 20 \sqrt{2}} - \sqrt{y - 20 \sqrt{2}} \cdot - 5 - 5 (- \sqrt{y - 20 \sqrt{2}}) - 5 \cdot - 5$

Step 2.8.3.1.3.1.2

Multiply $\sqrt{y - 20 \sqrt{2}}$ by $\sqrt{y - 20 \sqrt{2}}$ by adding the exponents.

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Step 2.8.3.1.3.1.2.1

Move $\sqrt{y - 20 \sqrt{2}}$ .

$x = - 1 \cdot - 1 (\sqrt{y - 20 \sqrt{2}} \sqrt{y - 20 \sqrt{2}}) - \sqrt{y - 20 \sqrt{2}} \cdot - 5 - 5 (- \sqrt{y - 20 \sqrt{2}}) - 5 \cdot - 5$

Step 2.8.3.1.3.1.2.2

Multiply $\sqrt{y - 20 \sqrt{2}}$ by $\sqrt{y - 20 \sqrt{2}}$ .

$x = - 1 \cdot - 1 {\sqrt{y - 20 \sqrt{2}}}^{2} - \sqrt{y - 20 \sqrt{2}} \cdot - 5 - 5 (- \sqrt{y - 20 \sqrt{2}}) - 5 \cdot - 5$

Step 2.8.3.1.3.1.3

Multiply $- 1$ by $- 1$ .

$x = 1 {\sqrt{y - 20 \sqrt{2}}}^{2} - \sqrt{y - 20 \sqrt{2}} \cdot - 5 - 5 (- \sqrt{y - 20 \sqrt{2}}) - 5 \cdot - 5$

Step 2.8.3.1.3.1.4

Multiply ${\sqrt{y - 20 \sqrt{2}}}^{2}$ by $1$ .

$x = {\sqrt{y - 20 \sqrt{2}}}^{2} - \sqrt{y - 20 \sqrt{2}} \cdot - 5 - 5 (- \sqrt{y - 20 \sqrt{2}}) - 5 \cdot - 5$

Step 2.8.3.1.3.1.5

Multiply $- 5$ by $- 1$ .

$x = {\sqrt{y - 20 \sqrt{2}}}^{2} + 5 \sqrt{y - 20 \sqrt{2}} - 5 (- \sqrt{y - 20 \sqrt{2}}) - 5 \cdot - 5$

Step 2.8.3.1.3.1.6

Multiply $- 1$ by $- 5$ .

$x = {\sqrt{y - 20 \sqrt{2}}}^{2} + 5 \sqrt{y - 20 \sqrt{2}} + 5 \sqrt{y - 20 \sqrt{2}} - 5 \cdot - 5$

Step 2.8.3.1.3.1.7

Multiply $- 5$ by $- 5$ .

$x = {\sqrt{y - 20 \sqrt{2}}}^{2} + 5 \sqrt{y - 20 \sqrt{2}} + 5 \sqrt{y - 20 \sqrt{2}} + 25$

Step 2.8.3.1.3.2

Add $5 \sqrt{y - 20 \sqrt{2}}$ and $5 \sqrt{y - 20 \sqrt{2}}$ .

$x = {\sqrt{y - 20 \sqrt{2}}}^{2} + 10 \sqrt{y - 20 \sqrt{2}} + 25$

Step 2.9

The complete solution is the result of both the positive and negative portions of the solution.

$x = {\sqrt{y - 20 \sqrt{2}}}^{2} - 10 \sqrt{y - 20 \sqrt{2}} + 25$

$x = {\sqrt{y - 20 \sqrt{2}}}^{2} + 10 \sqrt{y - 20 \sqrt{2}} + 25$

$x = {\sqrt{y - 20 \sqrt{2}}}^{2} - 10 \sqrt{y - 20 \sqrt{2}} + 25$

$x = {\sqrt{y - 20 \sqrt{2}}}^{2} + 10 \sqrt{y - 20 \sqrt{2}} + 25$