Algebra Examples

${(x - 5)}^{\frac{1}{2}} + 5 = 2$ ${(x - 5)}^{\frac{1}{2}} = - 3$ ${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 1

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

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Step 1.1

Subtract $5$ from both sides of the equation.

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

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 1.2

Subtract $5$ from $2$ .

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 3

Simplify the exponent.

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Step 3.1

Simplify the left side.

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Step 3.1.1

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

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Step 3.1.1.1

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

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Step 3.1.1.1.1

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

${(x - 5)}^{\frac{1}{2} \cdot 2} = {(- 3)}^{2}$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 3.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.1.1.1.2.1

Cancel the common factor.

${(x - 5)}^{\frac{1}{2} \cdot 2} = {(- 3)}^{2}$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 3.1.1.1.2.2

Rewrite the expression.

$x - 5 = {(- 3)}^{2}$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

$x - 5 = {(- 3)}^{2}$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

$x - 5 = {(- 3)}^{2}$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 3.1.1.2

Simplify.

$x - 5 = {(- 3)}^{2}$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

$x - 5 = {(- 3)}^{2}$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

$x - 5 = {(- 3)}^{2}$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Raise $- 3$ to the power of $2$ .

$x - 5 = 9$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

$x - 5 = 9$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

$x - 5 = 9$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 4

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

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Step 4.1

Add $5$ to both sides of the equation.

$x = 9 + 5$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 4.2

Add $9$ and $5$ .

$x = 14$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

$x = 14$

${(x - 5)}^{\frac{1}{2}} = - 3$

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 5

Exclude the solutions that do not make ${(x - 5)}^{\frac{1}{2}} + 5 = 2$ true.

No solution

Step 6