Algebra Examples

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

Step 1

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

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

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 $\sqrt{x^{2} - 4 x}$ as ${(x^{2} - 4 x)}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x^{2} - 4 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x^{2} - 4 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(- 4 + x)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(- 4 + x)}^{2}$ as $(- 4 + x) (- 4 + x)$ .

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

Step 2.3.1.2

Expand $(- 4 + x) (- 4 + x)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

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

Step 2.3.1.2.2

Apply the distributive property.

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

Step 2.3.1.2.3

Apply the distributive property.

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

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $- 4$ by $- 4$ .

$x^{2} - 4 x = 16 - 4 x + x \cdot - 4 + x \cdot x$

Step 2.3.1.3.1.2

Move $- 4$ to the left of $x$ .

$x^{2} - 4 x = 16 - 4 x - 4 \cdot x + x \cdot x$

Step 2.3.1.3.1.3

Multiply $x$ by $x$ .

$x^{2} - 4 x = 16 - 4 x - 4 x + x^{2}$

Step 2.3.1.3.2

Subtract $4 x$ from $- 4 x$ .

$x^{2} - 4 x = 16 - 8 x + x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Add $8 x$ to both sides of the equation.

$x^{2} - 4 x + 8 x = 16 + x^{2}$

Step 3.1.2

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

$x^{2} - 4 x + 8 x - x^{2} = 16$

Step 3.1.3

Combine the opposite terms in $x^{2} - 4 x + 8 x - x^{2}$ .

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Step 3.1.3.1

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

$- 4 x + 8 x + 0 = 16$

Step 3.1.3.2

Add $- 4 x + 8 x$ and $0$ .

$- 4 x + 8 x = 16$

Step 3.1.4

Add $- 4 x$ and $8 x$ .

$4 x = 16$

Step 3.2

Divide each term in $4 x = 16$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 x = 16$ by $4$ .

$\frac{4 x}{4} = \frac{16}{4}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{16}{4}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{16}{4}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $16$ by $4$ .

$x = 4$