# Algebra 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

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

Use the power rule to distribute the exponent.

Step 3.2.1.1.1

Apply the product rule to .

Step 3.2.1.1.2

Apply the product rule to .

Step 3.2.1.2

Raise to the power of .

Step 3.2.1.3

Multiply by .

Step 3.2.1.4

Raise to the power of .

Step 3.2.1.5

Raise to the power of .

Step 3.2.1.6

Cancel the common factor of .

Step 3.2.1.6.1

Factor out of .

Step 3.2.1.6.2

Cancel the common factor.

Step 3.2.1.6.3

Rewrite the expression.

Step 3.2.1.7

Multiply .

Step 3.2.1.7.1

Multiply by .

Step 3.2.1.7.2

Combine and .

Step 3.2.1.7.3

Multiply by .

Step 3.2.1.8

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

Reorder the factors of .

Step 3.2.2.7

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

Subtract from .

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