Algebra Examples

$y = \frac{1}{5} {(x - 3)}^{2} + 4$

Step 1

Simplify $\frac{1}{5} {(x - 3)}^{2} + 4$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$y = \frac{1}{5} ((x - 3) (x - 3)) + 4$

Step 1.1.2

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$y = \frac{1}{5} (x (x - 3) - 3 (x - 3)) + 4$

Step 1.1.2.2

Apply the distributive property.

$y = \frac{1}{5} (x \cdot x + x \cdot - 3 - 3 (x - 3)) + 4$

Step 1.1.2.3

Apply the distributive property.

$y = \frac{1}{5} (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) + 4$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

$y = \frac{1}{5} (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3) + 4$

Step 1.1.3.1.2

Move $- 3$ to the left of $x$ .

$y = \frac{1}{5} (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3) + 4$

Step 1.1.3.1.3

Multiply $- 3$ by $- 3$ .

$y = \frac{1}{5} (x^{2} - 3 x - 3 x + 9) + 4$

Step 1.1.3.2

Subtract $3 x$ from $- 3 x$ .

$y = \frac{1}{5} (x^{2} - 6 x + 9) + 4$

Step 1.1.4

Apply the distributive property.

$y = \frac{1}{5} x^{2} + \frac{1}{5} (- 6 x) + \frac{1}{5} \cdot 9 + 4$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Combine $\frac{1}{5}$ and $x^{2}$ .

$y = \frac{x^{2}}{5} + \frac{1}{5} (- 6 x) + \frac{1}{5} \cdot 9 + 4$

Step 1.1.5.2

Multiply $\frac{1}{5} (- 6 x)$ .

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Step 1.1.5.2.1

Combine $- 6$ and $\frac{1}{5}$ .

$y = \frac{x^{2}}{5} + \frac{- 6}{5} x + \frac{1}{5} \cdot 9 + 4$

Step 1.1.5.2.2

Combine $\frac{- 6}{5}$ and $x$ .

$y = \frac{x^{2}}{5} + \frac{- 6 x}{5} + \frac{1}{5} \cdot 9 + 4$

Step 1.1.5.3

Combine $\frac{1}{5}$ and $9$ .

$y = \frac{x^{2}}{5} + \frac{- 6 x}{5} + \frac{9}{5} + 4$

Step 1.1.6

Move the negative in front of the fraction.

$y = \frac{x^{2}}{5} - \frac{6 x}{5} + \frac{9}{5} + 4$

Step 1.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{x^{2}}{5} - \frac{6 x}{5} + \frac{9}{5} + 4 \cdot \frac{5}{5}$

Step 1.3

Combine $4$ and $\frac{5}{5}$ .

$y = \frac{x^{2}}{5} - \frac{6 x}{5} + \frac{9}{5} + \frac{4 \cdot 5}{5}$

Step 1.4

Combine the numerators over the common denominator.

$y = \frac{x^{2}}{5} - \frac{6 x}{5} + \frac{9 + 4 \cdot 5}{5}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $4$ by $5$ .

$y = \frac{x^{2}}{5} - \frac{6 x}{5} + \frac{9 + 20}{5}$

Step 1.5.2

Add $9$ and $20$ .

$y = \frac{x^{2}}{5} - \frac{6 x}{5} + \frac{29}{5}$

Step 2

Graph the equation in order to determine the intervals over which it is increasing or decreasing.

Increasing on: $(3, \infty)$

Decreasing on: $(- \infty, 3)$

Step 3