# Precalculus 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

Cancel the common factor of and .

Step 2.3.1.1

Factor out of .

Step 2.3.1.2

Cancel the common factors.

Step 2.3.1.2.1

Cancel the common factor.

Step 2.3.1.2.2

Rewrite the expression.

Step 2.3.2

Cancel the common factor of and .

Step 2.3.2.1

Factor out of .

Step 2.3.2.2

Cancel the common factors.

Step 2.3.2.2.1

Factor out of .

Step 2.3.2.2.2

Cancel the common factor.

Step 2.3.2.2.3

Rewrite the expression.

Step 2.3.2.2.4

Divide by .

Step 2.3.3

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

Raising to any positive power yields .

Step 3.2.1.2

Multiply by .

Step 3.2.2

Add and .

Step 3.2.3

The final answer is .

Step 4

Use the and values to find where the minimum occurs.

Step 5