# Finite Math Examples

Step 1

The minimum of a quadratic function occurs at . If is positive, the minimum value of the function is .

occurs at

Step 2

Step 2.1

Substitute in the values of and .

Step 2.2

Remove parentheses.

Step 2.3

Simplify .

Step 2.3.1

Multiply by .

Step 2.3.2

Move the negative in front of the fraction.

Step 2.3.3

Multiply .

Step 2.3.3.1

Multiply by .

Step 2.3.3.2

Multiply by .

Step 3

Step 3.1

Replace the variable with in the expression.

Step 3.2

Simplify the result.

Step 3.2.1

Simplify each term.

Step 3.2.1.1

Apply the product rule to .

Step 3.2.1.2

Raise to the power of .

Step 3.2.1.3

Raise to the power of .

Step 3.2.1.4

Multiply .

Step 3.2.1.4.1

Combine and .

Step 3.2.1.4.2

Multiply by .

Step 3.2.1.5

Move the negative in front of the fraction.

Step 3.2.2

Find the common denominator.

Step 3.2.2.1

Multiply by .

Step 3.2.2.2

Multiply by .

Step 3.2.2.3

Write as a fraction with denominator .

Step 3.2.2.4

Multiply by .

Step 3.2.2.5

Multiply by .

Step 3.2.2.6

Multiply by .

Step 3.2.3

Combine the numerators over the common denominator.

Step 3.2.4

Simplify each term.

Step 3.2.4.1

Multiply by .

Step 3.2.4.2

Multiply by .

Step 3.2.5

Simplify the expression.

Step 3.2.5.1

Subtract from .

Step 3.2.5.2

Add and .

Step 3.2.5.3

Move the negative in front of the fraction.

Step 3.2.6

The final answer is .

Step 4

Use the and values to find where the minimum occurs.

Step 5