Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.3.1.2

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

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

Step 1.3.1.3

Multiply $3$ by $3$ .

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

Step 1.3.2

Add $3 x$ and $3 x$ .

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

Step 1.4

Apply the distributive property.

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

Step 1.5

Simplify.

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Step 1.5.1

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

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

Step 1.5.2

Cancel the common factor of $2$ .

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Step 1.5.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 1.5.2.2

Factor $2$ out of $6 x$ .

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

Step 1.5.2.3

Cancel the common factor.

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

Step 1.5.2.4

Rewrite the expression.

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

Step 1.5.3

Multiply $3$ by $- 1$ .

$y \leq - \frac{x^{2}}{2} - 3 x - \frac{1}{2} \cdot 9 + 4$

Step 1.5.4

Multiply $- \frac{1}{2} \cdot 9$ .

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Step 1.5.4.1

Multiply $9$ by $- 1$ .

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

Step 1.5.4.2

Combine $- 9$ and $\frac{1}{2}$ .

$y \leq - \frac{x^{2}}{2} - 3 x + \frac{- 9}{2} + 4$

Step 1.6

Move the negative in front of the fraction.

$y \leq - \frac{x^{2}}{2} - 3 x - \frac{9}{2} + 4$

Step 2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y \leq - \frac{x^{2}}{2} - 3 x - \frac{9}{2} + 4 \cdot \frac{2}{2}$

Step 3

Combine $4$ and $\frac{2}{2}$ .

$y \leq - \frac{x^{2}}{2} - 3 x - \frac{9}{2} + \frac{4 \cdot 2}{2}$

Step 4

Combine the numerators over the common denominator.

$y \leq - \frac{x^{2}}{2} - 3 x + \frac{- 9 + 4 \cdot 2}{2}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $4$ by $2$ .

$y \leq - \frac{x^{2}}{2} - 3 x + \frac{- 9 + 8}{2}$

Step 5.2

Add $- 9$ and $8$ .

$y \leq - \frac{x^{2}}{2} - 3 x + \frac{- 1}{2}$

Step 6

Move the negative in front of the fraction.

$y \leq - \frac{x^{2}}{2} - 3 x - \frac{1}{2}$

Step 7