Algebra Examples

$\sqrt{{(x - 10)}^{2}} = x - 10$

Step 1

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

${\sqrt{{(x - 10)}^{2}}}^{2} = {(x - 10)}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(x - 10)}^{2}}$ as ${(x - 10)}^{\frac{2}{2}}$ .

${({(x - 10)}^{\frac{2}{2}})}^{2} = {(x - 10)}^{2}$

Step 2.2

Divide $2$ by $2$ .

${({(x - 10)}^{1})}^{2} = {(x - 10)}^{2}$

Step 2.3

Simplify the left side.

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Step 2.3.1

Multiply the exponents in ${({(x - 10)}^{1})}^{2}$ .

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Step 2.3.1.1

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

${(x - 10)}^{1 \cdot 2} = {(x - 10)}^{2}$

Step 2.3.1.2

Multiply $2$ by $1$ .

${(x - 10)}^{2} = {(x - 10)}^{2}$

Step 2.4

Simplify the right side.

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Step 2.4.1

Simplify ${(x - 10)}^{2}$ .

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Step 2.4.1.1

Rewrite ${(x - 10)}^{2}$ as $(x - 10) (x - 10)$ .

${(x - 10)}^{2} = (x - 10) (x - 10)$

Step 2.4.1.2

Expand $(x - 10) (x - 10)$ using the FOIL Method.

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Step 2.4.1.2.1

Apply the distributive property.

${(x - 10)}^{2} = x (x - 10) - 10 (x - 10)$

Step 2.4.1.2.2

Apply the distributive property.

${(x - 10)}^{2} = x \cdot x + x \cdot - 10 - 10 (x - 10)$

Step 2.4.1.2.3

Apply the distributive property.

${(x - 10)}^{2} = x \cdot x + x \cdot - 10 - 10 x - 10 \cdot - 10$

Step 2.4.1.3

Simplify and combine like terms.

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Step 2.4.1.3.1

Simplify each term.

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Step 2.4.1.3.1.1

Multiply $x$ by $x$ .

${(x - 10)}^{2} = x^{2} + x \cdot - 10 - 10 x - 10 \cdot - 10$

Step 2.4.1.3.1.2

Move $- 10$ to the left of $x$ .

${(x - 10)}^{2} = x^{2} - 10 \cdot x - 10 x - 10 \cdot - 10$

Step 2.4.1.3.1.3

Multiply $- 10$ by $- 10$ .

${(x - 10)}^{2} = x^{2} - 10 x - 10 x + 100$

Step 2.4.1.3.2

Subtract $10 x$ from $- 10 x$ .

${(x - 10)}^{2} = x^{2} - 20 x + 100$

Step 3

Solve for $x$ .

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Step 3.1

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

$x^{2} - 20 x + 100 = {(x - 10)}^{2}$

Step 3.2

Simplify ${(x - 10)}^{2}$ .

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Step 3.2.1

Rewrite.

$x^{2} - 20 x + 100 = 0 + 0 + {(x - 10)}^{2}$

Step 3.2.2

Rewrite ${(x - 10)}^{2}$ as $(x - 10) (x - 10)$ .

$x^{2} - 20 x + 100 = (x - 10) (x - 10)$

Step 3.2.3

Expand $(x - 10) (x - 10)$ using the FOIL Method.

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Step 3.2.3.1

Apply the distributive property.

$x^{2} - 20 x + 100 = x (x - 10) - 10 (x - 10)$

Step 3.2.3.2

Apply the distributive property.

$x^{2} - 20 x + 100 = x \cdot x + x \cdot - 10 - 10 (x - 10)$

Step 3.2.3.3

Apply the distributive property.

$x^{2} - 20 x + 100 = x \cdot x + x \cdot - 10 - 10 x - 10 \cdot - 10$

Step 3.2.4

Simplify and combine like terms.

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

Multiply $x$ by $x$ .

$x^{2} - 20 x + 100 = x^{2} + x \cdot - 10 - 10 x - 10 \cdot - 10$

Step 3.2.4.1.2

Move $- 10$ to the left of $x$ .

$x^{2} - 20 x + 100 = x^{2} - 10 \cdot x - 10 x - 10 \cdot - 10$

Step 3.2.4.1.3

Multiply $- 10$ by $- 10$ .

$x^{2} - 20 x + 100 = x^{2} - 10 x - 10 x + 100$

Step 3.2.4.2

Subtract $10 x$ from $- 10 x$ .

$x^{2} - 20 x + 100 = x^{2} - 20 x + 100$

Step 3.3

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

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Step 3.3.1

Subtract $x^{2}$ from both sides of the equation.

$x^{2} - 20 x + 100 - x^{2} = - 20 x + 100$

Step 3.3.2

Add $20 x$ to both sides of the equation.

$x^{2} - 20 x + 100 - x^{2} + 20 x = 100$

Step 3.3.3

Combine the opposite terms in $x^{2} - 20 x + 100 - x^{2} + 20 x$ .

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Step 3.3.3.1

Subtract $x^{2}$ from $x^{2}$ .

$- 20 x + 100 + 0 + 20 x = 100$

Step 3.3.3.2

Add $- 20 x + 100$ and $0$ .

$- 20 x + 100 + 20 x = 100$

Step 3.3.3.3

Add $- 20 x$ and $20 x$ .

$0 + 100 = 100$

Step 3.3.3.4

Add $0$ and $100$ .

$100 = 100$

Step 3.4

Since $100 = 100$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$