Algebra Examples

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

Step 1

Subtract $2$ from both sides of the equation.

$\frac{\sqrt{2 - 3 x}}{\sqrt{4 x}} - 2 = 0$

Step 2

Simplify $\frac{\sqrt{2 - 3 x}}{\sqrt{4 x}} - 2$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Simplify the denominator.

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Step 2.1.1.1

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2 - 3 x}}{\sqrt{2^{2} x}} - 2 = 0$

Step 2.1.1.2

Pull terms out from under the radical.

$\frac{\sqrt{2 - 3 x}}{2 \sqrt{x}} - 2 = 0$

Step 2.1.2

Multiply $\frac{\sqrt{2 - 3 x}}{2 \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{\sqrt{2 - 3 x}}{2 \sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}} - 2 = 0$

Step 2.1.3

Combine and simplify the denominator.

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Step 2.1.3.1

Multiply $\frac{\sqrt{2 - 3 x}}{2 \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{\sqrt{2 - 3 x} \sqrt{x}}{2 \sqrt{x} \sqrt{x}} - 2 = 0$

Step 2.1.3.2

Move $\sqrt{x}$ .

$\frac{\sqrt{2 - 3 x} \sqrt{x}}{2 (\sqrt{x} \sqrt{x})} - 2 = 0$

Step 2.1.3.3

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

$\frac{\sqrt{2 - 3 x} \sqrt{x}}{2 ({\sqrt{x}}^{1} \sqrt{x})} - 2 = 0$

Step 2.1.3.4

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

$\frac{\sqrt{2 - 3 x} \sqrt{x}}{2 ({\sqrt{x}}^{1} {\sqrt{x}}^{1})} - 2 = 0$

Step 2.1.3.5

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

$\frac{\sqrt{2 - 3 x} \sqrt{x}}{2 {\sqrt{x}}^{1 + 1}} - 2 = 0$

Step 2.1.3.6

Add $1$ and $1$ .

$\frac{\sqrt{2 - 3 x} \sqrt{x}}{2 {\sqrt{x}}^{2}} - 2 = 0$

Step 2.1.3.7

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

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Step 2.1.3.7.1

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

$\frac{\sqrt{2 - 3 x} \sqrt{x}}{2 {(x^{\frac{1}{2}})}^{2}} - 2 = 0$

Step 2.1.3.7.2

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

$\frac{\sqrt{2 - 3 x} \sqrt{x}}{2 x^{\frac{1}{2} \cdot 2}} - 2 = 0$

Step 2.1.3.7.3

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

$\frac{\sqrt{2 - 3 x} \sqrt{x}}{2 x^{\frac{2}{2}}} - 2 = 0$

Step 2.1.3.7.4

Cancel the common factor of $2$ .

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Step 2.1.3.7.4.1

Cancel the common factor.

$\frac{\sqrt{2 - 3 x} \sqrt{x}}{2 x^{\frac{2}{2}}} - 2 = 0$

Step 2.1.3.7.4.2

Rewrite the expression.

$\frac{\sqrt{2 - 3 x} \sqrt{x}}{2 x^{1}} - 2 = 0$

Step 2.1.3.7.5

Simplify.

$\frac{\sqrt{2 - 3 x} \sqrt{x}}{2 x} - 2 = 0$

Step 2.1.4

Combine using the product rule for radicals.

$\frac{\sqrt{(2 - 3 x) x}}{2 x} - 2 = 0$

Step 2.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

$\frac{\sqrt{(2 - 3 x) x}}{2 x} - 2 \cdot \frac{2 x}{2 x} = 0$

Step 2.3

Combine $- 2$ and $\frac{2 x}{2 x}$ .

$\frac{\sqrt{(2 - 3 x) x}}{2 x} + \frac{- 2 (2 x)}{2 x} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{\sqrt{(2 - 3 x) x} - 2 (2 x)}{2 x} = 0$

Step 2.5

Multiply $- 2$ by $2$ .

$\frac{\sqrt{(2 - 3 x) x} - 4 x}{2 x} = 0$

Step 2.6

Reorder factors in $\frac{\sqrt{(2 - 3 x) x} - 4 x}{2 x}$ .

$\frac{\sqrt{x (2 - 3 x)} - 4 x}{2 x} = 0$

Step 3

Set the numerator equal to zero.

$\sqrt{x (2 - 3 x)} - 4 x = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Add $4 x$ to both sides of the equation.

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

Step 4.2

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

${\sqrt{x (2 - 3 x)}}^{2} = {(4 x)}^{2}$

Step 4.3

Simplify each side of the equation.

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Step 4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x (2 - 3 x)}$ as ${(x (2 - 3 x))}^{\frac{1}{2}}$ .

${({(x (2 - 3 x))}^{\frac{1}{2}})}^{2} = {(4 x)}^{2}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Simplify ${({(x (2 - 3 x))}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.1.1

Simplify by multiplying through.

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Step 4.3.2.1.1.1

Multiply the exponents in ${({(x (2 - 3 x))}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.1.1.1.1

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

${(x (2 - 3 x))}^{\frac{1}{2} \cdot 2} = {(4 x)}^{2}$

Step 4.3.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 4.3.2.1.1.1.2.1

Cancel the common factor.

${(x (2 - 3 x))}^{\frac{1}{2} \cdot 2} = {(4 x)}^{2}$

Step 4.3.2.1.1.1.2.2

Rewrite the expression.

${(x (2 - 3 x))}^{1} = {(4 x)}^{2}$

Step 4.3.2.1.1.2

Apply the distributive property.

${(x \cdot 2 + x (- 3 x))}^{1} = {(4 x)}^{2}$

Step 4.3.2.1.1.3

Reorder.

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Step 4.3.2.1.1.3.1

Move $2$ to the left of $x$ .

${(2 \cdot x + x (- 3 x))}^{1} = {(4 x)}^{2}$

Step 4.3.2.1.1.3.2

Rewrite using the commutative property of multiplication.

${(2 \cdot x - 3 x \cdot x)}^{1} = {(4 x)}^{2}$

Step 4.3.2.1.2

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

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Step 4.3.2.1.2.1

Move $x$ .

${(2 x - 3 (x \cdot x))}^{1} = {(4 x)}^{2}$

Step 4.3.2.1.2.2

Multiply $x$ by $x$ .

${(2 x - 3 x^{2})}^{1} = {(4 x)}^{2}$

Step 4.3.2.1.3

Simplify.

$2 x - 3 x^{2} = {(4 x)}^{2}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

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

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Step 4.3.3.1.1

Apply the product rule to $4 x$ .

$2 x - 3 x^{2} = 4^{2} x^{2}$

Step 4.3.3.1.2

Raise $4$ to the power of $2$ .

$2 x - 3 x^{2} = 16 x^{2}$

Step 4.4

Solve for $x$ .

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Step 4.4.1

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

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Step 4.4.1.1

Subtract $16 x^{2}$ from both sides of the equation.

$2 x - 3 x^{2} - 16 x^{2} = 0$

Step 4.4.1.2

Subtract $16 x^{2}$ from $- 3 x^{2}$ .

$2 x - 19 x^{2} = 0$

Step 4.4.2

Factor the left side of the equation.

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Step 4.4.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$2 u - 19 u^{2} = 0$

Step 4.4.2.2

Factor $u$ out of $2 u - 19 u^{2}$ .

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Step 4.4.2.2.1

Factor $u$ out of $2 u$ .

$u \cdot 2 - 19 u^{2} = 0$

Step 4.4.2.2.2

Factor $u$ out of $- 19 u^{2}$ .

$u \cdot 2 + u (- 19 u) = 0$

Step 4.4.2.2.3

Factor $u$ out of $u \cdot 2 + u (- 19 u)$ .

$u (2 - 19 u) = 0$

Step 4.4.2.3

Replace all occurrences of $u$ with $x$ .

$x (2 - 19 x) = 0$

Step 4.4.3

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

$x = 0$

$2 - 19 x = 0$

Step 4.4.4

Set $x$ equal to $0$ .

$x = 0$

Step 4.4.5

Set $2 - 19 x$ equal to $0$ and solve for $x$ .

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Step 4.4.5.1

Set $2 - 19 x$ equal to $0$ .

$2 - 19 x = 0$

Step 4.4.5.2

Solve $2 - 19 x = 0$ for $x$ .

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Step 4.4.5.2.1

Subtract $2$ from both sides of the equation.

$- 19 x = - 2$

Step 4.4.5.2.2

Divide each term in $- 19 x = - 2$ by $- 19$ and simplify.

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Step 4.4.5.2.2.1

Divide each term in $- 19 x = - 2$ by $- 19$ .

$\frac{- 19 x}{- 19} = \frac{- 2}{- 19}$

Step 4.4.5.2.2.2

Simplify the left side.

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Step 4.4.5.2.2.2.1

Cancel the common factor of $- 19$ .

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Step 4.4.5.2.2.2.1.1

Cancel the common factor.

$\frac{- 19 x}{- 19} = \frac{- 2}{- 19}$

Step 4.4.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 19}$

Step 4.4.5.2.2.3

Simplify the right side.

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Step 4.4.5.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{2}{19}$

Step 4.4.6

The final solution is all the values that make $x (2 - 19 x) = 0$ true.

$x = 0, \frac{2}{19}$

Step 5

Exclude the solutions that do not make $\frac{\sqrt{x (2 - 3 x)} - 4 x}{2 x} = 0$ true.

$x = \frac{2}{19}$