Algebra Examples

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

Step 1

Simplify $\frac{1}{2} \cdot {(x - 6)}^{2} + 2$ .

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

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

Step 1.1.2

Expand $(x - 6) (x - 6)$ using the FOIL Method.

Tap for more steps...

Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

Step 1.1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.1.3.1

Simplify each term.

Tap for more steps...

Step 1.1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.1.3.1.2

Move $- 6$ to the left of $x$ .

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

Step 1.1.3.1.3

Multiply $- 6$ by $- 6$ .

$y \leq \frac{1}{2} \cdot (x^{2} - 6 x - 6 x + 36) + 2$

Step 1.1.3.2

Subtract $6 x$ from $- 6 x$ .

$y \leq \frac{1}{2} \cdot (x^{2} - 12 x + 36) + 2$

Step 1.1.4

Apply the distributive property.

$y \leq \frac{1}{2} x^{2} + \frac{1}{2} (- 12 x) + \frac{1}{2} \cdot 36 + 2$

Step 1.1.5

Simplify.

Tap for more steps...

Step 1.1.5.1

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

$y \leq \frac{x^{2}}{2} + \frac{1}{2} (- 12 x) + \frac{1}{2} \cdot 36 + 2$

Step 1.1.5.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.1.5.2.1

Factor $2$ out of $- 12 x$ .

$y \leq \frac{x^{2}}{2} + \frac{1}{2} (2 (- 6 x)) + \frac{1}{2} \cdot 36 + 2$

Step 1.1.5.2.2

Cancel the common factor.

$y \leq \frac{x^{2}}{2} + \frac{1}{2} (2 (- 6 x)) + \frac{1}{2} \cdot 36 + 2$

Step 1.1.5.2.3

Rewrite the expression.

$y \leq \frac{x^{2}}{2} - 6 x + \frac{1}{2} \cdot 36 + 2$

Step 1.1.5.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.1.5.3.1

Factor $2$ out of $36$ .

$y \leq \frac{x^{2}}{2} - 6 x + \frac{1}{2} \cdot (2 (18)) + 2$

Step 1.1.5.3.2

Cancel the common factor.

$y \leq \frac{x^{2}}{2} - 6 x + \frac{1}{2} \cdot (2 \cdot 18) + 2$

Step 1.1.5.3.3

Rewrite the expression.

$y \leq \frac{x^{2}}{2} - 6 x + 18 + 2$

Step 1.2

Add $18$ and $2$ .

$y \leq \frac{x^{2}}{2} - 6 x + 20$

Step 2

The equation is not linear, so a constant slope does not exist.

Not Linear

Step 3

Graph a solid line, then shade the area below the boundary line since $y$ is less than $\frac{x^{2}}{2} - 6 x + 20$ .

$y \leq \frac{x^{2}}{2} - 6 x + 20$

Step 4