Precalculus Examples

$- 6 y + \sqrt{x} = 4 x {(x + 2)}^{2}$

Step 1

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

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Step 1.1

Rewrite.

$- 6 y + \sqrt{x} = 0 + 0 + 4 x {(x + 2)}^{2}$

Step 1.2

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

$- 6 y + \sqrt{x} = 4 x ((x + 2) (x + 2))$

Step 1.3

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$- 6 y + \sqrt{x} = 4 x (x (x + 2) + 2 (x + 2))$

Step 1.3.2

Apply the distributive property.

$- 6 y + \sqrt{x} = 4 x (x \cdot x + x \cdot 2 + 2 (x + 2))$

Step 1.3.3

Apply the distributive property.

$- 6 y + \sqrt{x} = 4 x (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2)$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $x$ by $x$ .

$- 6 y + \sqrt{x} = 4 x (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2)$

Step 1.4.1.2

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

$- 6 y + \sqrt{x} = 4 x (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2)$

Step 1.4.1.3

Multiply $2$ by $2$ .

$- 6 y + \sqrt{x} = 4 x (x^{2} + 2 x + 2 x + 4)$

Step 1.4.2

Add $2 x$ and $2 x$ .

$- 6 y + \sqrt{x} = 4 x (x^{2} + 4 x + 4)$

Step 1.5

Apply the distributive property.

$- 6 y + \sqrt{x} = 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4$

Step 1.6

Simplify.

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Step 1.6.1

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

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Step 1.6.1.1

Move $x^{2}$ .

$- 6 y + \sqrt{x} = 4 (x^{2} x) + 4 x (4 x) + 4 x \cdot 4$

Step 1.6.1.2

Multiply $x^{2}$ by $x$ .

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Step 1.6.1.2.1

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

$- 6 y + \sqrt{x} = 4 (x^{2} x^{1}) + 4 x (4 x) + 4 x \cdot 4$

Step 1.6.1.2.2

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

$- 6 y + \sqrt{x} = 4 x^{2 + 1} + 4 x (4 x) + 4 x \cdot 4$

Step 1.6.1.3

Add $2$ and $1$ .

$- 6 y + \sqrt{x} = 4 x^{3} + 4 x (4 x) + 4 x \cdot 4$

Step 1.6.2

Rewrite using the commutative property of multiplication.

$- 6 y + \sqrt{x} = 4 x^{3} + 4 \cdot 4 x \cdot x + 4 x \cdot 4$

Step 1.6.3

Multiply $4$ by $4$ .

$- 6 y + \sqrt{x} = 4 x^{3} + 4 \cdot 4 x \cdot x + 16 x$

Step 1.7

Simplify each term.

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Step 1.7.1

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

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Step 1.7.1.1

Move $x$ .

$- 6 y + \sqrt{x} = 4 x^{3} + 4 \cdot 4 (x \cdot x) + 16 x$

Step 1.7.1.2

Multiply $x$ by $x$ .

$- 6 y + \sqrt{x} = 4 x^{3} + 4 \cdot 4 x^{2} + 16 x$

Step 1.7.2

Multiply $4$ by $4$ .

$- 6 y + \sqrt{x} = 4 x^{3} + 16 x^{2} + 16 x$

Step 2

Subtract $\sqrt{x}$ from both sides of the equation.

$- 6 y = 4 x^{3} + 16 x^{2} + 16 x - \sqrt{x}$

Step 3

Divide each term in $- 6 y = 4 x^{3} + 16 x^{2} + 16 x - \sqrt{x}$ by $- 6$ and simplify.

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Step 3.1

Divide each term in $- 6 y = 4 x^{3} + 16 x^{2} + 16 x - \sqrt{x}$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{4 x^{3}}{- 6} + \frac{16 x^{2}}{- 6} + \frac{16 x}{- 6} + \frac{- \sqrt{x}}{- 6}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 6$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{4 x^{3}}{- 6} + \frac{16 x^{2}}{- 6} + \frac{16 x}{- 6} + \frac{- \sqrt{x}}{- 6}$

Step 3.2.1.2

Divide $y$ by $1$ .

$y = \frac{4 x^{3}}{- 6} + \frac{16 x^{2}}{- 6} + \frac{16 x}{- 6} + \frac{- \sqrt{x}}{- 6}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Cancel the common factor of $4$ and $- 6$ .

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Step 3.3.1.1.1

Factor $2$ out of $4 x^{3}$ .

$y = \frac{2 (2 x^{3})}{- 6} + \frac{16 x^{2}}{- 6} + \frac{16 x}{- 6} + \frac{- \sqrt{x}}{- 6}$

Step 3.3.1.1.2

Cancel the common factors.

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Step 3.3.1.1.2.1

Factor $2$ out of $- 6$ .

$y = \frac{2 (2 x^{3})}{2 (- 3)} + \frac{16 x^{2}}{- 6} + \frac{16 x}{- 6} + \frac{- \sqrt{x}}{- 6}$

Step 3.3.1.1.2.2

Cancel the common factor.

$y = \frac{2 (2 x^{3})}{2 \cdot - 3} + \frac{16 x^{2}}{- 6} + \frac{16 x}{- 6} + \frac{- \sqrt{x}}{- 6}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{2 x^{3}}{- 3} + \frac{16 x^{2}}{- 6} + \frac{16 x}{- 6} + \frac{- \sqrt{x}}{- 6}$

Step 3.3.1.2

Move the negative in front of the fraction.

$y = - \frac{2 x^{3}}{3} + \frac{16 x^{2}}{- 6} + \frac{16 x}{- 6} + \frac{- \sqrt{x}}{- 6}$

Step 3.3.1.3

Cancel the common factor of $16$ and $- 6$ .

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Step 3.3.1.3.1

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

$y = - \frac{2 x^{3}}{3} + \frac{2 (8 x^{2})}{- 6} + \frac{16 x}{- 6} + \frac{- \sqrt{x}}{- 6}$

Step 3.3.1.3.2

Cancel the common factors.

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Step 3.3.1.3.2.1

Factor $2$ out of $- 6$ .

$y = - \frac{2 x^{3}}{3} + \frac{2 (8 x^{2})}{2 (- 3)} + \frac{16 x}{- 6} + \frac{- \sqrt{x}}{- 6}$

Step 3.3.1.3.2.2

Cancel the common factor.

$y = - \frac{2 x^{3}}{3} + \frac{2 (8 x^{2})}{2 \cdot - 3} + \frac{16 x}{- 6} + \frac{- \sqrt{x}}{- 6}$

Step 3.3.1.3.2.3

Rewrite the expression.

$y = - \frac{2 x^{3}}{3} + \frac{8 x^{2}}{- 3} + \frac{16 x}{- 6} + \frac{- \sqrt{x}}{- 6}$

Step 3.3.1.4

Move the negative in front of the fraction.

$y = - \frac{2 x^{3}}{3} - \frac{8 x^{2}}{3} + \frac{16 x}{- 6} + \frac{- \sqrt{x}}{- 6}$

Step 3.3.1.5

Cancel the common factor of $16$ and $- 6$ .

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Step 3.3.1.5.1

Factor $2$ out of $16 x$ .

$y = - \frac{2 x^{3}}{3} - \frac{8 x^{2}}{3} + \frac{2 (8 x)}{- 6} + \frac{- \sqrt{x}}{- 6}$

Step 3.3.1.5.2

Cancel the common factors.

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Step 3.3.1.5.2.1

Factor $2$ out of $- 6$ .

$y = - \frac{2 x^{3}}{3} - \frac{8 x^{2}}{3} + \frac{2 (8 x)}{2 (- 3)} + \frac{- \sqrt{x}}{- 6}$

Step 3.3.1.5.2.2

Cancel the common factor.

$y = - \frac{2 x^{3}}{3} - \frac{8 x^{2}}{3} + \frac{2 (8 x)}{2 \cdot - 3} + \frac{- \sqrt{x}}{- 6}$

Step 3.3.1.5.2.3

Rewrite the expression.

$y = - \frac{2 x^{3}}{3} - \frac{8 x^{2}}{3} + \frac{8 x}{- 3} + \frac{- \sqrt{x}}{- 6}$

Step 3.3.1.6

Move the negative in front of the fraction.

$y = - \frac{2 x^{3}}{3} - \frac{8 x^{2}}{3} - \frac{8 x}{3} + \frac{- \sqrt{x}}{- 6}$

Step 3.3.1.7

Dividing two negative values results in a positive value.

$y = - \frac{2 x^{3}}{3} - \frac{8 x^{2}}{3} - \frac{8 x}{3} + \frac{\sqrt{x}}{6}$

Step 4

To rewrite as a function of $x$ , write the equation so that $y$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

$f (x) = - \frac{2 x^{3}}{3} - \frac{8 x^{2}}{3} - \frac{8 x}{3} + \frac{\sqrt{x}}{6}$