Algebra Examples

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

Step 1

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

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

Step 2

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

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

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Rewrite ${(- 1 + \sqrt{x - 5})}^{2}$ as $(- 1 + \sqrt{x - 5}) (- 1 + \sqrt{x - 5})$ .

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

Step 3.3.1.2

Expand $(- 1 + \sqrt{x - 5}) (- 1 + \sqrt{x - 5})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

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

Step 3.3.1.2.2

Apply the distributive property.

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

Step 3.3.1.2.3

Apply the distributive property.

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

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 $- 1$ by $- 1$ .

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

Step 3.3.1.3.1.2

Rewrite $- 1 \sqrt{x - 5}$ as $- \sqrt{x - 5}$ .

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

Step 3.3.1.3.1.3

Move $- 1$ to the left of $\sqrt{x - 5}$ .

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

Step 3.3.1.3.1.4

Rewrite $- 1 \sqrt{x - 5}$ as $- \sqrt{x - 5}$ .

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

Step 3.3.1.3.1.5

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

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Step 3.3.1.3.1.5.1

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

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

Step 3.3.1.3.1.5.2

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

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

Step 3.3.1.3.1.5.3

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

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

Step 3.3.1.3.1.5.4

Add $1$ and $1$ .

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

Step 3.3.1.3.1.6

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

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Step 3.3.1.3.1.6.1

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

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

Step 3.3.1.3.1.6.2

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

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

Step 3.3.1.3.1.6.3

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

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

Step 3.3.1.3.1.6.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.6.4.1

Cancel the common factor.

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

Step 3.3.1.3.1.6.4.2

Rewrite the expression.

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

Step 3.3.1.3.1.6.5

Simplify.

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

Step 3.3.1.3.2

Subtract $5$ from $1$ .

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

Step 3.3.1.3.3

Subtract $\sqrt{x - 5}$ from $- \sqrt{x - 5}$ .

$x = - 4 - 2 \sqrt{x - 5} + x$

Step 4

Solve for $- 2 \sqrt{x - 5}$ .

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Step 4.1

Rewrite the equation as $- 4 - 2 \sqrt{x - 5} + x = x$ .

$- 4 - 2 \sqrt{x - 5} + x = x$

Step 4.2

Move all terms not containing $- 2 \sqrt{x - 5}$ to the right side of the equation.

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Step 4.2.1

Add $4$ to both sides of the equation.

$- 2 \sqrt{x - 5} + x = x + 4$

Step 4.2.2

Subtract $x$ from both sides of the equation.

$- 2 \sqrt{x - 5} = x + 4 - x$

Step 4.2.3

Combine the opposite terms in $x + 4 - x$ .

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Step 4.2.3.1

Subtract $x$ from $x$ .

$- 2 \sqrt{x - 5} = 0 + 4$

Step 4.2.3.2

Add $0$ and $4$ .

$- 2 \sqrt{x - 5} = 4$

Step 5

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

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

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

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

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

Step 6.2.1.2

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

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.

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

Step 6.2.1.3.2.2

Rewrite the expression.

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

Step 6.2.1.4

Simplify.

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

Step 6.2.1.5

Apply the distributive property.

$4 x + 4 \cdot - 5 = 4^{2}$

Step 6.2.1.6

Multiply $4$ by $- 5$ .

$4 x - 20 = 4^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

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

$4 x - 20 = 16$

Step 7

Solve for $x$ .

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Step 7.1

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

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Step 7.1.1

Add $20$ to both sides of the equation.

$4 x = 16 + 20$

Step 7.1.2

Add $16$ and $20$ .

$4 x = 36$

Step 7.2

Divide each term in $4 x = 36$ by $4$ and simplify.

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Step 7.2.1

Divide each term in $4 x = 36$ by $4$ .

$\frac{4 x}{4} = \frac{36}{4}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $4$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{36}{4}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{36}{4}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $36$ by $4$ .

$x = 9$

Step 8

Exclude the solutions that do not make $\sqrt{x} - \sqrt{x - 5} = - 1$ true.

$No solution$