Precalculus Examples

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

Step 1

Rewrite the equation as $\sqrt{\frac{x^{2} - 3 x + 2 x}{2 x - x}} = y$ .

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

Step 2

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

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

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 $\sqrt{\frac{x^{2} - 3 x + 2 x}{2 x - x}}$ as ${(\frac{x^{2} - 3 x + 2 x}{2 x - x})}^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify the numerator.

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Step 3.2.1.2.1

Add $- 3 x$ and $2 x$ .

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

Step 3.2.1.2.2

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

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Step 3.2.1.2.2.1

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

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

Step 3.2.1.2.2.2

Factor $x$ out of $- x$ .

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

Step 3.2.1.2.2.3

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

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

Step 3.2.1.3

Simplify terms.

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Step 3.2.1.3.1

Subtract $x$ from $2 x$ .

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

Step 3.2.1.3.2

Cancel the common factor of $x$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

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

Step 3.2.1.3.2.2

Divide $x - 1$ by $1$ .

${(x - 1)}^{1} = y^{2}$

Step 3.2.1.3.3

Simplify.

$x - 1 = y^{2}$

Step 4

Add $1$ to both sides of the equation.

$x = y^{2} + 1$