Algebra Examples

$\sqrt{x - 4} - 2 = \sqrt{x} - 3$

Step 1

Move all terms not containing $\sqrt{x - 4}$ to the right side of the equation.

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Step 1.1

Add $2$ to both sides of the equation.

$\sqrt{x - 4} = \sqrt{x} - 3 + 2$

Step 1.2

Add $- 3$ and $2$ .

$\sqrt{x - 4} = \sqrt{x} - 1$

Step 2

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

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

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

$x - 4 = (\sqrt{x} - 1) (\sqrt{x} - 1)$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Apply the distributive property.

$x - 4 = \sqrt{x} (\sqrt{x} - 1) - 1 (\sqrt{x} - 1)$

Step 3.3.1.2.2

Apply the distributive property.

$x - 4 = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 1 - 1 (\sqrt{x} - 1)$

Step 3.3.1.2.3

Apply the distributive property.

$x - 4 = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 1 - 1 \sqrt{x} - 1 \cdot - 1$

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 $\sqrt{x} \sqrt{x}$ .

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Step 3.3.1.3.1.1.1

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

$x - 4 = {\sqrt{x}}^{1} \sqrt{x} + \sqrt{x} \cdot - 1 - 1 \sqrt{x} - 1 \cdot - 1$

Step 3.3.1.3.1.1.2

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

$x - 4 = {\sqrt{x}}^{1} {\sqrt{x}}^{1} + \sqrt{x} \cdot - 1 - 1 \sqrt{x} - 1 \cdot - 1$

Step 3.3.1.3.1.1.3

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

$x - 4 = {\sqrt{x}}^{1 + 1} + \sqrt{x} \cdot - 1 - 1 \sqrt{x} - 1 \cdot - 1$

Step 3.3.1.3.1.1.4

Add $1$ and $1$ .

$x - 4 = {\sqrt{x}}^{2} + \sqrt{x} \cdot - 1 - 1 \sqrt{x} - 1 \cdot - 1$

Step 3.3.1.3.1.2

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

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Step 3.3.1.3.1.2.1

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

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

Step 3.3.1.3.1.2.2

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

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

Step 3.3.1.3.1.2.3

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

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

Step 3.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.2.4.1

Cancel the common factor.

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

Step 3.3.1.3.1.2.4.2

Rewrite the expression.

$x - 4 = x^{1} + \sqrt{x} \cdot - 1 - 1 \sqrt{x} - 1 \cdot - 1$

Step 3.3.1.3.1.2.5

Simplify.

$x - 4 = x + \sqrt{x} \cdot - 1 - 1 \sqrt{x} - 1 \cdot - 1$

Step 3.3.1.3.1.3

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

$x - 4 = x - 1 \cdot \sqrt{x} - 1 \sqrt{x} - 1 \cdot - 1$

Step 3.3.1.3.1.4

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

$x - 4 = x - \sqrt{x} - 1 \sqrt{x} - 1 \cdot - 1$

Step 3.3.1.3.1.5

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

$x - 4 = x - \sqrt{x} - \sqrt{x} - 1 \cdot - 1$

Step 3.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

$x - 4 = x - \sqrt{x} - \sqrt{x} + 1$

Step 3.3.1.3.2

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

$x - 4 = x - 2 \sqrt{x} + 1$

Step 4

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

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Step 4.1

Rewrite the equation as $x - 2 \sqrt{x} + 1 = x - 4$ .

$x - 2 \sqrt{x} + 1 = x - 4$

Step 4.2

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

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Step 4.2.1

Subtract $x$ from both sides of the equation.

$- 2 \sqrt{x} + 1 = x - 4 - x$

Step 4.2.2

Subtract $1$ from both sides of the equation.

$- 2 \sqrt{x} = x - 4 - x - 1$

Step 4.2.3

Combine the opposite terms in $x - 4 - x - 1$ .

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Step 4.2.3.1

Subtract $x$ from $x$ .

$- 2 \sqrt{x} = 0 - 4 - 1$

Step 4.2.3.2

Subtract $4$ from $0$ .

$- 2 \sqrt{x} = - 4 - 1$

Step 4.2.4

Subtract $1$ from $- 4$ .

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

Step 5

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

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

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

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

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

Step 6.2.1.2

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

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

Step 6.2.1.3

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

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Step 6.2.1.3.1

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

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

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.

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

Step 6.2.1.3.2.2

Rewrite the expression.

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

Step 6.2.1.4

Simplify.

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Raise $- 5$ to the power of $2$ .

$4 x = 25$

Step 7

Divide each term in $4 x = 25$ by $4$ and simplify.

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Step 7.1

Divide each term in $4 x = 25$ by $4$ .

$\frac{4 x}{4} = \frac{25}{4}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $4$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{25}{4}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{25}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{25}{4}$

Decimal Form:

$x = 6.25$

Mixed Number Form:

$x = 6 \frac{1}{4}$