Calculus Examples

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

Step 1

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

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

Step 2

Simplify each side of the equation.

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

$3 x y = {(2 + x^{2} y)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(2 + x^{2} y)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(2 + x^{2} y)}^{2}$ as $(2 + x^{2} y) (2 + x^{2} y)$ .

$3 x y = (2 + x^{2} y) (2 + x^{2} y)$

Step 2.3.1.2

Expand $(2 + x^{2} y) (2 + x^{2} y)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$3 x y = 2 (2 + x^{2} y) + x^{2} y (2 + x^{2} y)$

Step 2.3.1.2.2

Apply the distributive property.

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

Step 2.3.1.2.3

Apply the distributive property.

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

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $2$ by $2$ .

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

Step 2.3.1.3.1.2

Move $2$ to the left of $x^{2} y$ .

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

Step 2.3.1.3.1.3

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.3.1.3.1.3.1

Move $x^{2}$ .

$3 x y = 4 + 2 x^{2} y + 2 x^{2} y + x^{2} x^{2} y \cdot y$

Step 2.3.1.3.1.3.2

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

$3 x y = 4 + 2 x^{2} y + 2 x^{2} y + x^{2 + 2} y \cdot y$

Step 2.3.1.3.1.3.3

Add $2$ and $2$ .

$3 x y = 4 + 2 x^{2} y + 2 x^{2} y + x^{4} y \cdot y$

Step 2.3.1.3.1.4

Multiply $y$ by $y$ by adding the exponents.

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Step 2.3.1.3.1.4.1

Move $y$ .

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

Step 2.3.1.3.1.4.2

Multiply $y$ by $y$ .

$3 x y = 4 + 2 x^{2} y + 2 x^{2} y + x^{4} y^{2}$

Step 2.3.1.3.2

Add $2 x^{2} y$ and $2 x^{2} y$ .

$3 x y = 4 + 4 x^{2} y + x^{4} y^{2}$

Step 3

Solve for $y$ .

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Step 3.1

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

$4 + 4 x^{2} y + x^{4} y^{2} = 3 x y$

Step 3.2

Subtract $3 x y$ from both sides of the equation.

$4 + 4 x^{2} y + x^{4} y^{2} - 3 x y = 0$

Step 3.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.4

Substitute the values $a = x^{4}$ , $b = 4 x^{2} - 3 x$ , and $c = 4$ into the quadratic formula and solve for $y$ .

$\frac{- (4 x^{2} - 3 x) \pm \sqrt{{(4 x^{2} - 3 x)}^{2} - 4 \cdot (x^{4} \cdot 4)}}{2 x^{4}}$

Step 3.5

Simplify.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Apply the distributive property.

$y = \frac{- (4 x^{2}) - (- 3 x) \pm \sqrt{{(4 x^{2} - 3 x)}^{2} - 4 \cdot x^{4} \cdot 4}}{2 x^{4}}$

Step 3.5.1.2

Multiply $4$ by $- 1$ .

$y = \frac{- 4 x^{2} - (- 3 x) \pm \sqrt{{(4 x^{2} - 3 x)}^{2} - 4 \cdot x^{4} \cdot 4}}{2 x^{4}}$

Step 3.5.1.3

Multiply $- 3$ by $- 1$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{{(4 x^{2} - 3 x)}^{2} - 4 \cdot x^{4} \cdot 4}}{2 x^{4}}$

Step 3.5.1.4

Rewrite $4 \cdot x^{4} \cdot 4$ as ${(2 \cdot x^{2} \cdot 2)}^{2}$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{{(4 x^{2} - 3 x)}^{2} - {(2 \cdot x^{2} \cdot 2)}^{2}}}{2 x^{4}}$

Step 3.5.1.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4 x^{2} - 3 x$ and $b = 2 \cdot x^{2} \cdot 2$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{(4 x^{2} - 3 x + 2 \cdot x^{2} \cdot 2) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.5.1.6

Simplify.

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Step 3.5.1.6.1

Factor $x$ out of $4 x^{2} - 3 x + 2 \cdot x^{2} \cdot 2$ .

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Step 3.5.1.6.1.1

Factor $x$ out of $4 x^{2}$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{(x (4 x) - 3 x + 2 \cdot x^{2} \cdot 2) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.5.1.6.1.2

Factor $x$ out of $- 3 x$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{(x (4 x) + x \cdot - 3 + 2 \cdot x^{2} \cdot 2) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.5.1.6.1.3

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

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{(x (4 x) + x \cdot - 3 + x (2 \cdot x \cdot 2)) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.5.1.6.1.4

Factor $x$ out of $x (4 x) + x \cdot - 3$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{(x (4 x - 3) + x (2 \cdot x \cdot 2)) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.5.1.6.1.5

Factor $x$ out of $x (4 x - 3) + x (2 \cdot x \cdot 2)$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (4 x - 3 + 2 \cdot x \cdot 2) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.5.1.6.2

Multiply $2$ by $2$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (4 x - 3 + 4 x) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.5.1.6.3

Add $4 x$ and $4 x$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.5.1.6.4

Combine exponents.

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Step 3.5.1.6.4.1

Multiply $2$ by $2$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) (4 x^{2} - 3 x - (4 \cdot x^{2}))}}{2 x^{4}}$

Step 3.5.1.6.4.2

Multiply $4$ by $- 1$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) (4 x^{2} - 3 x - 4 x^{2})}}{2 x^{4}}$

Step 3.5.1.7

Combine the opposite terms in $4 x^{2} - 3 x - 4 x^{2}$ .

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Step 3.5.1.7.1

Subtract $4 x^{2}$ from $4 x^{2}$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) (- 3 x + 0)}}{2 x^{4}}$

Step 3.5.1.7.2

Add $- 3 x$ and $0$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) (- 3 x)}}{2 x^{4}}$

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) \cdot (- 3 x)}}{2 x^{4}}$

Step 3.5.1.8

Combine exponents.

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Step 3.5.1.8.1

Raise $x$ to the power of $1$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x \cdot x (8 x - 3) \cdot - 3}}{2 x^{4}}$

Step 3.5.1.8.2

Raise $x$ to the power of $1$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x \cdot x (8 x - 3) \cdot - 3}}{2 x^{4}}$

Step 3.5.1.8.3

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

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x^{1 + 1} (8 x - 3) \cdot - 3}}{2 x^{4}}$

Step 3.5.1.8.4

Add $1$ and $1$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x^{2} (8 x - 3) \cdot - 3}}{2 x^{4}}$

Step 3.5.1.9

Add parentheses.

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x^{2} ((8 x - 3) \cdot - 3)}}{2 x^{4}}$

Step 3.5.1.10

Pull terms out from under the radical.

$y = \frac{- 4 x^{2} + 3 x \pm x \sqrt{(8 x - 3) \cdot - 3}}{2 x^{4}}$

Step 3.5.2

Reorder factors in $\frac{- 4 x^{2} + 3 x \pm x \sqrt{(8 x - 3) \cdot - 3}}{2 x^{4}}$ .

$y = \frac{- 4 x^{2} + 3 x \pm x \sqrt{- 3 (8 x - 3)}}{2 x^{4}}$

Step 3.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Apply the distributive property.

$y = \frac{- (4 x^{2}) - (- 3 x) \pm \sqrt{{(4 x^{2} - 3 x)}^{2} - 4 \cdot x^{4} \cdot 4}}{2 x^{4}}$

Step 3.6.1.2

Multiply $4$ by $- 1$ .

$y = \frac{- 4 x^{2} - (- 3 x) \pm \sqrt{{(4 x^{2} - 3 x)}^{2} - 4 \cdot x^{4} \cdot 4}}{2 x^{4}}$

Step 3.6.1.3

Multiply $- 3$ by $- 1$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{{(4 x^{2} - 3 x)}^{2} - 4 \cdot x^{4} \cdot 4}}{2 x^{4}}$

Step 3.6.1.4

Rewrite $4 \cdot x^{4} \cdot 4$ as ${(2 \cdot x^{2} \cdot 2)}^{2}$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{{(4 x^{2} - 3 x)}^{2} - {(2 \cdot x^{2} \cdot 2)}^{2}}}{2 x^{4}}$

Step 3.6.1.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4 x^{2} - 3 x$ and $b = 2 \cdot x^{2} \cdot 2$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{(4 x^{2} - 3 x + 2 \cdot x^{2} \cdot 2) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.6.1.6

Simplify.

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Step 3.6.1.6.1

Factor $x$ out of $4 x^{2} - 3 x + 2 \cdot x^{2} \cdot 2$ .

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Step 3.6.1.6.1.1

Factor $x$ out of $4 x^{2}$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{(x (4 x) - 3 x + 2 \cdot x^{2} \cdot 2) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.6.1.6.1.2

Factor $x$ out of $- 3 x$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{(x (4 x) + x \cdot - 3 + 2 \cdot x^{2} \cdot 2) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.6.1.6.1.3

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

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{(x (4 x) + x \cdot - 3 + x (2 \cdot x \cdot 2)) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.6.1.6.1.4

Factor $x$ out of $x (4 x) + x \cdot - 3$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{(x (4 x - 3) + x (2 \cdot x \cdot 2)) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.6.1.6.1.5

Factor $x$ out of $x (4 x - 3) + x (2 \cdot x \cdot 2)$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (4 x - 3 + 2 \cdot x \cdot 2) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.6.1.6.2

Multiply $2$ by $2$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (4 x - 3 + 4 x) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.6.1.6.3

Add $4 x$ and $4 x$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.6.1.6.4

Combine exponents.

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Step 3.6.1.6.4.1

Multiply $2$ by $2$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) (4 x^{2} - 3 x - (4 \cdot x^{2}))}}{2 x^{4}}$

Step 3.6.1.6.4.2

Multiply $4$ by $- 1$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) (4 x^{2} - 3 x - 4 x^{2})}}{2 x^{4}}$

Step 3.6.1.7

Combine the opposite terms in $4 x^{2} - 3 x - 4 x^{2}$ .

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Step 3.6.1.7.1

Subtract $4 x^{2}$ from $4 x^{2}$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) (- 3 x + 0)}}{2 x^{4}}$

Step 3.6.1.7.2

Add $- 3 x$ and $0$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) (- 3 x)}}{2 x^{4}}$

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) \cdot (- 3 x)}}{2 x^{4}}$

Step 3.6.1.8

Combine exponents.

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Step 3.6.1.8.1

Raise $x$ to the power of $1$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x \cdot x (8 x - 3) \cdot - 3}}{2 x^{4}}$

Step 3.6.1.8.2

Raise $x$ to the power of $1$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x \cdot x (8 x - 3) \cdot - 3}}{2 x^{4}}$

Step 3.6.1.8.3

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

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x^{1 + 1} (8 x - 3) \cdot - 3}}{2 x^{4}}$

Step 3.6.1.8.4

Add $1$ and $1$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x^{2} (8 x - 3) \cdot - 3}}{2 x^{4}}$

Step 3.6.1.9

Add parentheses.

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x^{2} ((8 x - 3) \cdot - 3)}}{2 x^{4}}$

Step 3.6.1.10

Pull terms out from under the radical.

$y = \frac{- 4 x^{2} + 3 x \pm x \sqrt{(8 x - 3) \cdot - 3}}{2 x^{4}}$

Step 3.6.2

Reorder factors in $\frac{- 4 x^{2} + 3 x \pm x \sqrt{(8 x - 3) \cdot - 3}}{2 x^{4}}$ .

$y = \frac{- 4 x^{2} + 3 x \pm x \sqrt{- 3 (8 x - 3)}}{2 x^{4}}$

Step 3.6.3

Change the $\pm$ to $+$ .

$y = \frac{- 4 x^{2} + 3 x + x \sqrt{- 3 (8 x - 3)}}{2 x^{4}}$

Step 3.6.4

Factor $x$ out of $- 4 x^{2} + 3 x + x \sqrt{- 3 (8 x - 3)}$ .

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Step 3.6.4.1

Factor $x$ out of $- 4 x^{2}$ .

$y = \frac{x (- 4 x) + 3 x + x \sqrt{- 3 (8 x - 3)}}{2 x^{4}}$

Step 3.6.4.2

Factor $x$ out of $3 x$ .

$y = \frac{x (- 4 x) + x \cdot 3 + x \sqrt{- 3 (8 x - 3)}}{2 x^{4}}$

Step 3.6.4.3

Factor $x$ out of $x \sqrt{- 3 (8 x - 3)}$ .

$y = \frac{x (- 4 x) + x \cdot 3 + x (\sqrt{- 3 (8 x - 3)})}{2 x^{4}}$

Step 3.6.4.4

Factor $x$ out of $x (- 4 x) + x \cdot 3$ .

$y = \frac{x (- 4 x + 3) + x (\sqrt{- 3 (8 x - 3)})}{2 x^{4}}$

Step 3.6.4.5

Factor $x$ out of $x (- 4 x + 3) + x (\sqrt{- 3 (8 x - 3)})$ .

$y = \frac{x (- 4 x + 3 + \sqrt{- 3 (8 x - 3)})}{2 x^{4}}$

Step 3.6.5

Cancel the common factors.

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Step 3.6.5.1

Factor $x$ out of $2 x^{4}$ .

$y = \frac{x (- 4 x + 3 + \sqrt{- 3 (8 x - 3)})}{x (2 x^{3})}$

Step 3.6.5.2

Cancel the common factor.

$y = \frac{x (- 4 x + 3 + \sqrt{- 3 (8 x - 3)})}{x (2 x^{3})}$

Step 3.6.5.3

Rewrite the expression.

$y = \frac{- 4 x + 3 + \sqrt{- 3 (8 x - 3)}}{2 x^{3}}$

Step 3.6.6

Factor $- 1$ out of $- 4 x$ .

$y = \frac{- (4 x) + 3 + \sqrt{- 3 (8 x - 3)}}{2 x^{3}}$

Step 3.6.7

Rewrite $3$ as $- 1 (- 3)$ .

$y = \frac{- (4 x) - 1 \cdot - 3 + \sqrt{- 3 (8 x - 3)}}{2 x^{3}}$

Step 3.6.8

Factor $- 1$ out of $- (4 x) - 1 (- 3)$ .

$y = \frac{- (4 x - 3) + \sqrt{- 3 (8 x - 3)}}{2 x^{3}}$

Step 3.6.9

Factor $- 1$ out of $\sqrt{- 3 (8 x - 3)}$ .

$y = \frac{- (4 x - 3) - 1 (- \sqrt{- 3 (8 x - 3)})}{2 x^{3}}$

Step 3.6.10

Factor $- 1$ out of $- (4 x - 3) - 1 (- \sqrt{- 3 (8 x - 3)})$ .

$y = \frac{- (4 x - 3 - \sqrt{- 3 (8 x - 3)})}{2 x^{3}}$

Step 3.6.11

Rewrite $- (4 x - 3 - \sqrt{- 3 (8 x - 3)})$ as $- 1 (4 x - 3 - \sqrt{- 3 (8 x - 3)})$ .

$y = \frac{- 1 (4 x - 3 - \sqrt{- 3 (8 x - 3)})}{2 x^{3}}$

Step 3.6.12

Move the negative in front of the fraction.

$y = - \frac{4 x - 3 - \sqrt{- 3 (8 x - 3)}}{2 x^{3}}$

Step 3.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

Apply the distributive property.

$y = \frac{- (4 x^{2}) - (- 3 x) \pm \sqrt{{(4 x^{2} - 3 x)}^{2} - 4 \cdot x^{4} \cdot 4}}{2 x^{4}}$

Step 3.7.1.2

Multiply $4$ by $- 1$ .

$y = \frac{- 4 x^{2} - (- 3 x) \pm \sqrt{{(4 x^{2} - 3 x)}^{2} - 4 \cdot x^{4} \cdot 4}}{2 x^{4}}$

Step 3.7.1.3

Multiply $- 3$ by $- 1$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{{(4 x^{2} - 3 x)}^{2} - 4 \cdot x^{4} \cdot 4}}{2 x^{4}}$

Step 3.7.1.4

Rewrite $4 \cdot x^{4} \cdot 4$ as ${(2 \cdot x^{2} \cdot 2)}^{2}$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{{(4 x^{2} - 3 x)}^{2} - {(2 \cdot x^{2} \cdot 2)}^{2}}}{2 x^{4}}$

Step 3.7.1.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4 x^{2} - 3 x$ and $b = 2 \cdot x^{2} \cdot 2$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{(4 x^{2} - 3 x + 2 \cdot x^{2} \cdot 2) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.7.1.6

Simplify.

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Step 3.7.1.6.1

Factor $x$ out of $4 x^{2} - 3 x + 2 \cdot x^{2} \cdot 2$ .

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Step 3.7.1.6.1.1

Factor $x$ out of $4 x^{2}$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{(x (4 x) - 3 x + 2 \cdot x^{2} \cdot 2) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.7.1.6.1.2

Factor $x$ out of $- 3 x$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{(x (4 x) + x \cdot - 3 + 2 \cdot x^{2} \cdot 2) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.7.1.6.1.3

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

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{(x (4 x) + x \cdot - 3 + x (2 \cdot x \cdot 2)) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.7.1.6.1.4

Factor $x$ out of $x (4 x) + x \cdot - 3$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{(x (4 x - 3) + x (2 \cdot x \cdot 2)) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.7.1.6.1.5

Factor $x$ out of $x (4 x - 3) + x (2 \cdot x \cdot 2)$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (4 x - 3 + 2 \cdot x \cdot 2) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.7.1.6.2

Multiply $2$ by $2$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (4 x - 3 + 4 x) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.7.1.6.3

Add $4 x$ and $4 x$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) (4 x^{2} - 3 x - (2 \cdot x^{2} \cdot 2))}}{2 x^{4}}$

Step 3.7.1.6.4

Combine exponents.

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Step 3.7.1.6.4.1

Multiply $2$ by $2$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) (4 x^{2} - 3 x - (4 \cdot x^{2}))}}{2 x^{4}}$

Step 3.7.1.6.4.2

Multiply $4$ by $- 1$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) (4 x^{2} - 3 x - 4 x^{2})}}{2 x^{4}}$

Step 3.7.1.7

Combine the opposite terms in $4 x^{2} - 3 x - 4 x^{2}$ .

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Step 3.7.1.7.1

Subtract $4 x^{2}$ from $4 x^{2}$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) (- 3 x + 0)}}{2 x^{4}}$

Step 3.7.1.7.2

Add $- 3 x$ and $0$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) (- 3 x)}}{2 x^{4}}$

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x (8 x - 3) \cdot (- 3 x)}}{2 x^{4}}$

Step 3.7.1.8

Combine exponents.

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Step 3.7.1.8.1

Raise $x$ to the power of $1$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x \cdot x (8 x - 3) \cdot - 3}}{2 x^{4}}$

Step 3.7.1.8.2

Raise $x$ to the power of $1$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x \cdot x (8 x - 3) \cdot - 3}}{2 x^{4}}$

Step 3.7.1.8.3

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

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x^{1 + 1} (8 x - 3) \cdot - 3}}{2 x^{4}}$

Step 3.7.1.8.4

Add $1$ and $1$ .

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x^{2} (8 x - 3) \cdot - 3}}{2 x^{4}}$

Step 3.7.1.9

Add parentheses.

$y = \frac{- 4 x^{2} + 3 x \pm \sqrt{x^{2} ((8 x - 3) \cdot - 3)}}{2 x^{4}}$

Step 3.7.1.10

Pull terms out from under the radical.

$y = \frac{- 4 x^{2} + 3 x \pm x \sqrt{(8 x - 3) \cdot - 3}}{2 x^{4}}$

Step 3.7.2

Reorder factors in $\frac{- 4 x^{2} + 3 x \pm x \sqrt{(8 x - 3) \cdot - 3}}{2 x^{4}}$ .

$y = \frac{- 4 x^{2} + 3 x \pm x \sqrt{- 3 (8 x - 3)}}{2 x^{4}}$

Step 3.7.3

Change the $\pm$ to $-$ .

$y = \frac{- 4 x^{2} + 3 x - x \sqrt{- 3 (8 x - 3)}}{2 x^{4}}$

Step 3.7.4

Factor $x$ out of $- 4 x^{2} + 3 x - x \sqrt{- 3 (8 x - 3)}$ .

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Step 3.7.4.1

Factor $x$ out of $- 4 x^{2}$ .

$y = \frac{x (- 4 x) + 3 x - x \sqrt{- 3 (8 x - 3)}}{2 x^{4}}$

Step 3.7.4.2

Factor $x$ out of $3 x$ .

$y = \frac{x (- 4 x) + x \cdot 3 - x \sqrt{- 3 (8 x - 3)}}{2 x^{4}}$

Step 3.7.4.3

Factor $x$ out of $- x \sqrt{- 3 (8 x - 3)}$ .

$y = \frac{x (- 4 x) + x \cdot 3 + x (- \sqrt{- 3 (8 x - 3)})}{2 x^{4}}$

Step 3.7.4.4

Factor $x$ out of $x (- 4 x) + x \cdot 3$ .

$y = \frac{x (- 4 x + 3) + x (- \sqrt{- 3 (8 x - 3)})}{2 x^{4}}$

Step 3.7.4.5

Factor $x$ out of $x (- 4 x + 3) + x (- \sqrt{- 3 (8 x - 3)})$ .

$y = \frac{x (- 4 x + 3 - \sqrt{- 3 (8 x - 3)})}{2 x^{4}}$

Step 3.7.5

Cancel the common factors.

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Step 3.7.5.1

Factor $x$ out of $2 x^{4}$ .

$y = \frac{x (- 4 x + 3 - \sqrt{- 3 (8 x - 3)})}{x (2 x^{3})}$

Step 3.7.5.2

Cancel the common factor.

$y = \frac{x (- 4 x + 3 - \sqrt{- 3 (8 x - 3)})}{x (2 x^{3})}$

Step 3.7.5.3

Rewrite the expression.

$y = \frac{- 4 x + 3 - \sqrt{- 3 (8 x - 3)}}{2 x^{3}}$

Step 3.7.6

Factor $- 1$ out of $- 4 x$ .

$y = \frac{- (4 x) + 3 - \sqrt{- 3 (8 x - 3)}}{2 x^{3}}$

Step 3.7.7

Rewrite $3$ as $- 1 (- 3)$ .

$y = \frac{- (4 x) - 1 \cdot - 3 - \sqrt{- 3 (8 x - 3)}}{2 x^{3}}$

Step 3.7.8

Factor $- 1$ out of $- (4 x) - 1 (- 3)$ .

$y = \frac{- (4 x - 3) - \sqrt{- 3 (8 x - 3)}}{2 x^{3}}$

Step 3.7.9

Factor $- 1$ out of $- \sqrt{- 3 (8 x - 3)}$ .

$y = \frac{- (4 x - 3) - (\sqrt{- 3 (8 x - 3)})}{2 x^{3}}$

Step 3.7.10

Factor $- 1$ out of $- (4 x - 3) - (\sqrt{- 3 (8 x - 3)})$ .

$y = \frac{- (4 x - 3 + \sqrt{- 3 (8 x - 3)})}{2 x^{3}}$

Step 3.7.11

Rewrite $- (4 x - 3 + \sqrt{- 3 (8 x - 3)})$ as $- 1 (4 x - 3 + \sqrt{- 3 (8 x - 3)})$ .

$y = \frac{- 1 (4 x - 3 + \sqrt{- 3 (8 x - 3)})}{2 x^{3}}$

Step 3.7.12

Move the negative in front of the fraction.

$y = - \frac{4 x - 3 + \sqrt{- 3 (8 x - 3)}}{2 x^{3}}$

Step 3.8

The final answer is the combination of both solutions.

$y = - \frac{4 x - 3 - \sqrt{- 3 (8 x - 3)}}{2 x^{3}}$

$y = - \frac{4 x - 3 + \sqrt{- 3 (8 x - 3)}}{2 x^{3}}$

$y = - \frac{4 x - 3 - \sqrt{- 3 (8 x - 3)}}{2 x^{3}}$

$y = - \frac{4 x - 3 + \sqrt{- 3 (8 x - 3)}}{2 x^{3}}$