Algebra Examples

$h (x) = \frac{1}{2} (3 - x) (2 - x)$

Step 1

Rewrite the function as an equation.

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

Step 2

Remove parentheses.

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

Step 3

Simplify $\frac{1}{2} \cdot ((3 - x) (2 - x))$ .

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Step 3.1

Expand $(3 - x) (2 - x)$ using the FOIL Method.

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Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

Apply the distributive property.

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

Step 3.1.3

Apply the distributive property.

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

Step 3.2

Simplify and combine like terms.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $3$ by $2$ .

$y = \frac{1}{2} \cdot (6 + 3 (- x) - x \cdot 2 - x (- x))$

Step 3.2.1.2

Multiply $- 1$ by $3$ .

$y = \frac{1}{2} \cdot (6 - 3 x - x \cdot 2 - x (- x))$

Step 3.2.1.3

Multiply $2$ by $- 1$ .

$y = \frac{1}{2} \cdot (6 - 3 x - 2 x - x (- x))$

Step 3.2.1.4

Rewrite using the commutative property of multiplication.

$y = \frac{1}{2} \cdot (6 - 3 x - 2 x - 1 \cdot - 1 x \cdot x)$

Step 3.2.1.5

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

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Step 3.2.1.5.1

Move $x$ .

$y = \frac{1}{2} \cdot (6 - 3 x - 2 x - 1 \cdot - 1 (x \cdot x))$

Step 3.2.1.5.2

Multiply $x$ by $x$ .

$y = \frac{1}{2} \cdot (6 - 3 x - 2 x - 1 \cdot - 1 x^{2})$

Step 3.2.1.6

Multiply $- 1$ by $- 1$ .

$y = \frac{1}{2} \cdot (6 - 3 x - 2 x + 1 x^{2})$

Step 3.2.1.7

Multiply $x^{2}$ by $1$ .

$y = \frac{1}{2} \cdot (6 - 3 x - 2 x + x^{2})$

Step 3.2.2

Subtract $2 x$ from $- 3 x$ .

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Simplify.

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Step 3.4.1

Cancel the common factor of $2$ .

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Step 3.4.1.1

Factor $2$ out of $6$ .

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

Step 3.4.1.2

Cancel the common factor.

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

Step 3.4.1.3

Rewrite the expression.

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

Step 3.4.2

Multiply $\frac{1}{2} (- 5 x)$ .

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Step 3.4.2.1

Combine $- 5$ and $\frac{1}{2}$ .

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

Step 3.4.2.2

Combine $\frac{- 5}{2}$ and $x$ .

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

Step 3.4.3

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

$y = 3 + \frac{- 5 x}{2} + \frac{x^{2}}{2}$

Step 3.5

Move the negative in front of the fraction.

$y = 3 - \frac{5 x}{2} + \frac{x^{2}}{2}$

Step 3.6

Move $3$ .

$y = - \frac{5 x}{2} + \frac{x^{2}}{2} + 3$

Step 3.7

Reorder $- \frac{5 x}{2}$ and $\frac{x^{2}}{2}$ .

$y = \frac{x^{2}}{2} - \frac{5 x}{2} + 3$