Algebra Examples

$\sqrt{x} - 3 = \sqrt{x - 9}$

Step 1

Add $3$ to both sides of the equation.

$\sqrt{x} = \sqrt{x - 9} + 3$

Step 2

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

${\sqrt{x}}^{2} = {(\sqrt{x - 9} + 3)}^{2}$

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

${(x^{\frac{1}{2}})}^{2} = {(\sqrt{x - 9} + 3)}^{2}$

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

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{x - 9} + 3)}^{2}$

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.

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{x - 9} + 3)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(\sqrt{x - 9} + 3)}^{2}$

Step 3.2.1.2

Simplify.

$x = {(\sqrt{x - 9} + 3)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(\sqrt{x - 9} + 3)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(\sqrt{x - 9} + 3)}^{2}$ as $(\sqrt{x - 9} + 3) (\sqrt{x - 9} + 3)$ .

$x = (\sqrt{x - 9} + 3) (\sqrt{x - 9} + 3)$

Step 3.3.1.2

Expand $(\sqrt{x - 9} + 3) (\sqrt{x - 9} + 3)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$x = \sqrt{x - 9} (\sqrt{x - 9} + 3) + 3 (\sqrt{x - 9} + 3)$

Step 3.3.1.2.2

Apply the distributive property.

$x = \sqrt{x - 9} \sqrt{x - 9} + \sqrt{x - 9} \cdot 3 + 3 (\sqrt{x - 9} + 3)$

Step 3.3.1.2.3

Apply the distributive property.

$x = \sqrt{x - 9} \sqrt{x - 9} + \sqrt{x - 9} \cdot 3 + 3 \sqrt{x - 9} + 3 \cdot 3$

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 $\sqrt{x - 9} \sqrt{x - 9}$ .

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Step 3.3.1.3.1.1.1

Raise $\sqrt{x - 9}$ to the power of $1$ .

$x = {\sqrt{x - 9}}^{1} \sqrt{x - 9} + \sqrt{x - 9} \cdot 3 + 3 \sqrt{x - 9} + 3 \cdot 3$

Step 3.3.1.3.1.1.2

Raise $\sqrt{x - 9}$ to the power of $1$ .

$x = {\sqrt{x - 9}}^{1} {\sqrt{x - 9}}^{1} + \sqrt{x - 9} \cdot 3 + 3 \sqrt{x - 9} + 3 \cdot 3$

Step 3.3.1.3.1.1.3

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

$x = {\sqrt{x - 9}}^{1 + 1} + \sqrt{x - 9} \cdot 3 + 3 \sqrt{x - 9} + 3 \cdot 3$

Step 3.3.1.3.1.1.4

Add $1$ and $1$ .

$x = {\sqrt{x - 9}}^{2} + \sqrt{x - 9} \cdot 3 + 3 \sqrt{x - 9} + 3 \cdot 3$

Step 3.3.1.3.1.2

Rewrite ${\sqrt{x - 9}}^{2}$ as $x - 9$ .

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Step 3.3.1.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 9}$ as ${(x - 9)}^{\frac{1}{2}}$ .

$x = {({(x - 9)}^{\frac{1}{2}})}^{2} + \sqrt{x - 9} \cdot 3 + 3 \sqrt{x - 9} + 3 \cdot 3$

Step 3.3.1.3.1.2.2

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

$x = {(x - 9)}^{\frac{1}{2} \cdot 2} + \sqrt{x - 9} \cdot 3 + 3 \sqrt{x - 9} + 3 \cdot 3$

Step 3.3.1.3.1.2.3

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

$x = {(x - 9)}^{\frac{2}{2}} + \sqrt{x - 9} \cdot 3 + 3 \sqrt{x - 9} + 3 \cdot 3$

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.

$x = {(x - 9)}^{\frac{2}{2}} + \sqrt{x - 9} \cdot 3 + 3 \sqrt{x - 9} + 3 \cdot 3$

Step 3.3.1.3.1.2.4.2

Rewrite the expression.

$x = {(x - 9)}^{1} + \sqrt{x - 9} \cdot 3 + 3 \sqrt{x - 9} + 3 \cdot 3$

Step 3.3.1.3.1.2.5

Simplify.

$x = x - 9 + \sqrt{x - 9} \cdot 3 + 3 \sqrt{x - 9} + 3 \cdot 3$

Step 3.3.1.3.1.3

Move $3$ to the left of $\sqrt{x - 9}$ .

$x = x - 9 + 3 \cdot \sqrt{x - 9} + 3 \sqrt{x - 9} + 3 \cdot 3$

Step 3.3.1.3.1.4

Multiply $3$ by $3$ .

$x = x - 9 + 3 \sqrt{x - 9} + 3 \sqrt{x - 9} + 9$

Step 3.3.1.3.2

Add $- 9$ and $9$ .

$x = x + 0 + 3 \sqrt{x - 9} + 3 \sqrt{x - 9}$

Step 3.3.1.3.3

Add $x$ and $0$ .

$x = x + 3 \sqrt{x - 9} + 3 \sqrt{x - 9}$

Step 3.3.1.3.4

Add $3 \sqrt{x - 9}$ and $3 \sqrt{x - 9}$ .

$x = x + 6 \sqrt{x - 9}$

Step 4

Solve for $6 \sqrt{x - 9}$ .

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Step 4.1

Rewrite the equation as $x + 6 \sqrt{x - 9} = x$ .

$x + 6 \sqrt{x - 9} = x$

Step 4.2

Move all terms not containing $6 \sqrt{x - 9}$ to the right side of the equation.

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Step 4.2.1

Subtract $x$ from both sides of the equation.

$6 \sqrt{x - 9} = x - x$

Step 4.2.2

Subtract $x$ from $x$ .

$6 \sqrt{x - 9} = 0$

Step 5

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

${(6 \sqrt{x - 9})}^{2} = 0^{2}$

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 $\sqrt{x - 9}$ as ${(x - 9)}^{\frac{1}{2}}$ .

${(6 {(x - 9)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(6 {(x - 9)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $6 {(x - 9)}^{\frac{1}{2}}$ .

$6^{2} {({(x - 9)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.2.1.2

Raise $6$ to the power of $2$ .

$36 {({(x - 9)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.2.1.3

Multiply the exponents in ${({(x - 9)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.3.1

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

$36 {(x - 9)}^{\frac{1}{2} \cdot 2} = 0^{2}$

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.

$36 {(x - 9)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$36 {(x - 9)}^{1} = 0^{2}$

Step 6.2.1.4

Simplify.

$36 (x - 9) = 0^{2}$

Step 6.2.1.5

Apply the distributive property.

$36 x + 36 \cdot - 9 = 0^{2}$

Step 6.2.1.6

Multiply $36$ by $- 9$ .

$36 x - 324 = 0^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Raising $0$ to any positive power yields $0$ .

$36 x - 324 = 0$

Step 7

Solve for $x$ .

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Step 7.1

Add $324$ to both sides of the equation.

$36 x = 324$

Step 7.2

Divide each term in $36 x = 324$ by $36$ and simplify.

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Step 7.2.1

Divide each term in $36 x = 324$ by $36$ .

$\frac{36 x}{36} = \frac{324}{36}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $36$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{36 x}{36} = \frac{324}{36}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{324}{36}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $324$ by $36$ .

$x = 9$