Algebra Examples

$\sqrt{2 x} - \sqrt{36 - 2 x} = 6$

Step 1

Solve for $\sqrt{2 x}$ .

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Step 1.1

Factor $2$ out of $36 - 2 x$ .

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Step 1.1.1

Factor $2$ out of $36$ .

$\sqrt{2 x} - \sqrt{2 (18) - 2 x} = 6$

Step 1.1.2

Factor $2$ out of $- 2 x$ .

$\sqrt{2 x} - \sqrt{2 (18) + 2 (- x)} = 6$

Step 1.1.3

Factor $2$ out of $2 (18) + 2 (- x)$ .

$\sqrt{2 x} - \sqrt{2 (18 - x)} = 6$

Step 1.2

Add $\sqrt{2 (18 - x)}$ to both sides of the equation.

$\sqrt{2 x} = 6 + \sqrt{2 (18 - x)}$

Step 2

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

${\sqrt{2 x}}^{2} = {(6 + \sqrt{2 (18 - x)})}^{2}$

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

${({(2 x)}^{\frac{1}{2}})}^{2} = {(6 + \sqrt{2 (18 - x)})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

${(2 x)}^{\frac{1}{2} \cdot 2} = {(6 + \sqrt{2 (18 - x)})}^{2}$

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.

${(2 x)}^{\frac{1}{2} \cdot 2} = {(6 + \sqrt{2 (18 - x)})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 x)}^{1} = {(6 + \sqrt{2 (18 - x)})}^{2}$

Step 3.2.1.2

Simplify.

$2 x = {(6 + \sqrt{2 (18 - x)})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(6 + \sqrt{2 (18 - x)})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(6 + \sqrt{2 (18 - x)})}^{2}$ as $(6 + \sqrt{2 (18 - x)}) (6 + \sqrt{2 (18 - x)})$ .

$2 x = (6 + \sqrt{2 (18 - x)}) (6 + \sqrt{2 (18 - x)})$

Step 3.3.1.2

Expand $(6 + \sqrt{2 (18 - x)}) (6 + \sqrt{2 (18 - x)})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$2 x = 6 (6 + \sqrt{2 (18 - x)}) + \sqrt{2 (18 - x)} (6 + \sqrt{2 (18 - x)})$

Step 3.3.1.2.2

Apply the distributive property.

$2 x = 6 \cdot 6 + 6 \sqrt{2 (18 - x)} + \sqrt{2 (18 - x)} (6 + \sqrt{2 (18 - x)})$

Step 3.3.1.2.3

Apply the distributive property.

$2 x = 6 \cdot 6 + 6 \sqrt{2 (18 - x)} + \sqrt{2 (18 - x)} \cdot 6 + \sqrt{2 (18 - x)} \sqrt{2 (18 - x)}$

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 $6$ by $6$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + \sqrt{2 (18 - x)} \cdot 6 + \sqrt{2 (18 - x)} \sqrt{2 (18 - x)}$

Step 3.3.1.3.1.2

Move $6$ to the left of $\sqrt{2 (18 - x)}$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \cdot \sqrt{2 (18 - x)} + \sqrt{2 (18 - x)} \sqrt{2 (18 - x)}$

Step 3.3.1.3.1.3

Multiply $\sqrt{2 (18 - x)} \sqrt{2 (18 - x)}$ .

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Step 3.3.1.3.1.3.1

Raise $\sqrt{2 (18 - x)}$ to the power of $1$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {\sqrt{2 (18 - x)}}^{1} \sqrt{2 (18 - x)}$

Step 3.3.1.3.1.3.2

Raise $\sqrt{2 (18 - x)}$ to the power of $1$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {\sqrt{2 (18 - x)}}^{1} {\sqrt{2 (18 - x)}}^{1}$

Step 3.3.1.3.1.3.3

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

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {\sqrt{2 (18 - x)}}^{1 + 1}$

Step 3.3.1.3.1.3.4

Add $1$ and $1$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {\sqrt{2 (18 - x)}}^{2}$

Step 3.3.1.3.1.4

Rewrite ${\sqrt{2 (18 - x)}}^{2}$ as $2 (18 - x)$ .

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Step 3.3.1.3.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (18 - x)}$ as ${(2 (18 - x))}^{\frac{1}{2}}$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {({(2 (18 - x))}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.4.2

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

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {(2 (18 - x))}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.4.3

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

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {(2 (18 - x))}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.4.4.1

Cancel the common factor.

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {(2 (18 - x))}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4.2

Rewrite the expression.

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {(2 (18 - x))}^{1}$

Step 3.3.1.3.1.4.5

Simplify.

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + 2 (18 - x)$

Step 3.3.1.3.1.5

Apply the distributive property.

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + 2 \cdot 18 + 2 (- x)$

Step 3.3.1.3.1.6

Multiply $2$ by $18$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + 36 + 2 (- x)$

Step 3.3.1.3.1.7

Multiply $- 1$ by $2$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + 36 - 2 x$

Step 3.3.1.3.2

Add $36$ and $36$ .

$2 x = 72 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} - 2 x$

Step 3.3.1.3.3

Add $6 \sqrt{2 (18 - x)}$ and $6 \sqrt{2 (18 - x)}$ .

$2 x = 72 + 12 \sqrt{2 (18 - x)} - 2 x$

Step 4

Solve for $12 \sqrt{2 (18 - x)}$ .

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Step 4.1

Rewrite the equation as $72 + 12 \sqrt{2 (18 - x)} - 2 x = 2 x$ .

$72 + 12 \sqrt{2 (18 - x)} - 2 x = 2 x$

Step 4.2

Move all terms not containing $12 \sqrt{2 (18 - x)}$ to the right side of the equation.

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Step 4.2.1

Subtract $72$ from both sides of the equation.

$12 \sqrt{2 (18 - x)} - 2 x = 2 x - 72$

Step 4.2.2

Add $2 x$ to both sides of the equation.

$12 \sqrt{2 (18 - x)} = 2 x - 72 + 2 x$

Step 4.2.3

Add $2 x$ and $2 x$ .

$12 \sqrt{2 (18 - x)} = 4 x - 72$

Step 5

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

${(12 \sqrt{2 (18 - x)})}^{2} = {(4 x - 72)}^{2}$

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 $\sqrt{2 (18 - x)}$ as ${(2 (18 - x))}^{\frac{1}{2}}$ .

${(12 {(2 (18 - x))}^{\frac{1}{2}})}^{2} = {(4 x - 72)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(12 {(2 (18 - x))}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the distributive property.

${(12 {(2 \cdot 18 + 2 (- x))}^{\frac{1}{2}})}^{2} = {(4 x - 72)}^{2}$

Step 6.2.1.2

Simplify the expression.

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Step 6.2.1.2.1

Multiply $2$ by $18$ .

${(12 {(36 + 2 (- x))}^{\frac{1}{2}})}^{2} = {(4 x - 72)}^{2}$

Step 6.2.1.2.2

Multiply $- 1$ by $2$ .

${(12 {(36 - 2 x)}^{\frac{1}{2}})}^{2} = {(4 x - 72)}^{2}$

Step 6.2.1.2.3

Apply the product rule to $12 {(36 - 2 x)}^{\frac{1}{2}}$ .

$12^{2} {({(36 - 2 x)}^{\frac{1}{2}})}^{2} = {(4 x - 72)}^{2}$

Step 6.2.1.2.4

Raise $12$ to the power of $2$ .

$144 {({(36 - 2 x)}^{\frac{1}{2}})}^{2} = {(4 x - 72)}^{2}$

Step 6.2.1.2.5

Multiply the exponents in ${({(36 - 2 x)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.2.5.1

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

$144 {(36 - 2 x)}^{\frac{1}{2} \cdot 2} = {(4 x - 72)}^{2}$

Step 6.2.1.2.5.2

Cancel the common factor of $2$ .

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Step 6.2.1.2.5.2.1

Cancel the common factor.

$144 {(36 - 2 x)}^{\frac{1}{2} \cdot 2} = {(4 x - 72)}^{2}$

Step 6.2.1.2.5.2.2

Rewrite the expression.

$144 {(36 - 2 x)}^{1} = {(4 x - 72)}^{2}$

Step 6.2.1.3

Simplify.

$144 (36 - 2 x) = {(4 x - 72)}^{2}$

Step 6.2.1.4

Apply the distributive property.

$144 \cdot 36 + 144 (- 2 x) = {(4 x - 72)}^{2}$

Step 6.2.1.5

Multiply.

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Step 6.2.1.5.1

Multiply $144$ by $36$ .

$5184 + 144 (- 2 x) = {(4 x - 72)}^{2}$

Step 6.2.1.5.2

Multiply $- 2$ by $144$ .

$5184 - 288 x = {(4 x - 72)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(4 x - 72)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(4 x - 72)}^{2}$ as $(4 x - 72) (4 x - 72)$ .

$5184 - 288 x = (4 x - 72) (4 x - 72)$

Step 6.3.1.2

Expand $(4 x - 72) (4 x - 72)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$5184 - 288 x = 4 x (4 x - 72) - 72 (4 x - 72)$

Step 6.3.1.2.2

Apply the distributive property.

$5184 - 288 x = 4 x (4 x) + 4 x \cdot - 72 - 72 (4 x - 72)$

Step 6.3.1.2.3

Apply the distributive property.

$5184 - 288 x = 4 x (4 x) + 4 x \cdot - 72 - 72 (4 x) - 72 \cdot - 72$

Step 6.3.1.3

Simplify and combine like terms.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$5184 - 288 x = 4 \cdot 4 x \cdot x + 4 x \cdot - 72 - 72 (4 x) - 72 \cdot - 72$

Step 6.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 6.3.1.3.1.2.1

Move $x$ .

$5184 - 288 x = 4 \cdot 4 (x \cdot x) + 4 x \cdot - 72 - 72 (4 x) - 72 \cdot - 72$

Step 6.3.1.3.1.2.2

Multiply $x$ by $x$ .

$5184 - 288 x = 4 \cdot 4 x^{2} + 4 x \cdot - 72 - 72 (4 x) - 72 \cdot - 72$

Step 6.3.1.3.1.3

Multiply $4$ by $4$ .

$5184 - 288 x = 16 x^{2} + 4 x \cdot - 72 - 72 (4 x) - 72 \cdot - 72$

Step 6.3.1.3.1.4

Multiply $- 72$ by $4$ .

$5184 - 288 x = 16 x^{2} - 288 x - 72 (4 x) - 72 \cdot - 72$

Step 6.3.1.3.1.5

Multiply $4$ by $- 72$ .

$5184 - 288 x = 16 x^{2} - 288 x - 288 x - 72 \cdot - 72$

Step 6.3.1.3.1.6

Multiply $- 72$ by $- 72$ .

$5184 - 288 x = 16 x^{2} - 288 x - 288 x + 5184$

Step 6.3.1.3.2

Subtract $288 x$ from $- 288 x$ .

$5184 - 288 x = 16 x^{2} - 576 x + 5184$

Step 7

Solve for $x$ .

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Step 7.1

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

$16 x^{2} - 576 x + 5184 = 5184 - 288 x$

Step 7.2

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

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Step 7.2.1

Add $288 x$ to both sides of the equation.

$16 x^{2} - 576 x + 5184 + 288 x = 5184$

Step 7.2.2

Add $- 576 x$ and $288 x$ .

$16 x^{2} - 288 x + 5184 = 5184$

Step 7.3

Subtract $5184$ from both sides of the equation.

$16 x^{2} - 288 x + 5184 - 5184 = 0$

Step 7.4

Combine the opposite terms in $16 x^{2} - 288 x + 5184 - 5184$ .

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Step 7.4.1

Subtract $5184$ from $5184$ .

$16 x^{2} - 288 x + 0 = 0$

Step 7.4.2

Add $16 x^{2} - 288 x$ and $0$ .

$16 x^{2} - 288 x = 0$

Step 7.5

Factor $16 x$ out of $16 x^{2} - 288 x$ .

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Step 7.5.1

Factor $16 x$ out of $16 x^{2}$ .

$16 x (x) - 288 x = 0$

Step 7.5.2

Factor $16 x$ out of $- 288 x$ .

$16 x (x) + 16 x (- 18) = 0$

Step 7.5.3

Factor $16 x$ out of $16 x (x) + 16 x (- 18)$ .

$16 x (x - 18) = 0$

Step 7.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 18 = 0$

Step 7.7

Set $x$ equal to $0$ .

$x = 0$

Step 7.8

Set $x - 18$ equal to $0$ and solve for $x$ .

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Step 7.8.1

Set $x - 18$ equal to $0$ .

$x - 18 = 0$

Step 7.8.2

Add $18$ to both sides of the equation.

$x = 18$

Step 7.9

The final solution is all the values that make $16 x (x - 18) = 0$ true.

$x = 0, 18$

Step 8

Exclude the solutions that do not make $\sqrt{2 x} - \sqrt{36 - 2 x} = 6$ true.

$x = 18$

Step 9