Algebra Examples

$f (x) = 2 x^{2} - 10 x + 6$

Step 1

The minimum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is positive, the minimum value of the function is $f (- \frac{b}{2 a})$ .

$f_{min}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 2

Find the value of $x = - \frac{b}{2 a}$ .

Tap for more steps...

Step 2.1

Substitute in the values of $a$ and $b$ .

$x = - \frac{- 10}{2 (2)}$

Step 2.2

Remove parentheses.

$x = - \frac{- 10}{2 (2)}$

Step 2.3

Simplify $- \frac{- 10}{2 (2)}$ .

Tap for more steps...

Step 2.3.1

Cancel the common factor of $- 10$ and $2$ .

Tap for more steps...

Step 2.3.1.1

Factor $2$ out of $- 10$ .

$x = - \frac{2 \cdot - 5}{2 \cdot 2}$

Step 2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.3.1.2.1

Factor $2$ out of $2 \cdot 2$ .

$x = - \frac{2 \cdot - 5}{2 (2)}$

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot - 5}{2 \cdot 2}$

Step 2.3.1.2.3

Rewrite the expression.

$x = - \frac{- 5}{2}$

Step 2.3.2

Move the negative in front of the fraction.

$x = - - \frac{5}{2}$

Step 2.3.3

Multiply $- - \frac{5}{2}$ .

Tap for more steps...

Step 2.3.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{5}{2})$

Step 2.3.3.2

Multiply $\frac{5}{2}$ by $1$ .

$x = \frac{5}{2}$

Step 3

Evaluate $f (\frac{5}{2})$ .

Tap for more steps...

Step 3.1

Replace the variable $x$ with $\frac{5}{2}$ in the expression.

$f (\frac{5}{2}) = 2 {(\frac{5}{2})}^{2} - 10 (\frac{5}{2}) + 6$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Apply the product rule to $\frac{5}{2}$ .

$f (\frac{5}{2}) = 2 (\frac{5^{2}}{2^{2}}) - 10 (\frac{5}{2}) + 6$

Step 3.2.1.2

Raise $5$ to the power of $2$ .

$f (\frac{5}{2}) = 2 (\frac{25}{2^{2}}) - 10 (\frac{5}{2}) + 6$

Step 3.2.1.3

Raise $2$ to the power of $2$ .

$f (\frac{5}{2}) = 2 (\frac{25}{4}) - 10 (\frac{5}{2}) + 6$

Step 3.2.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.4.1

Factor $2$ out of $4$ .

$f (\frac{5}{2}) = 2 (\frac{25}{2 (2)}) - 10 (\frac{5}{2}) + 6$

Step 3.2.1.4.2

Cancel the common factor.

$f (\frac{5}{2}) = 2 (\frac{25}{2 \cdot 2}) - 10 (\frac{5}{2}) + 6$

Step 3.2.1.4.3

Rewrite the expression.

$f (\frac{5}{2}) = \frac{25}{2} - 10 (\frac{5}{2}) + 6$

Step 3.2.1.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.5.1

Factor $2$ out of $- 10$ .

$f (\frac{5}{2}) = \frac{25}{2} + 2 (- 5) (\frac{5}{2}) + 6$

Step 3.2.1.5.2

Cancel the common factor.

$f (\frac{5}{2}) = \frac{25}{2} + 2 \cdot (- 5 (\frac{5}{2})) + 6$

Step 3.2.1.5.3

Rewrite the expression.

$f (\frac{5}{2}) = \frac{25}{2} - 5 \cdot 5 + 6$

Step 3.2.1.6

Multiply $- 5$ by $5$ .

$f (\frac{5}{2}) = \frac{25}{2} - 25 + 6$

Step 3.2.2

Find the common denominator.

Tap for more steps...

Step 3.2.2.1

Write $- 25$ as a fraction with denominator $1$ .

$f (\frac{5}{2}) = \frac{25}{2} + \frac{- 25}{1} + 6$

Step 3.2.2.2

Multiply $\frac{- 25}{1}$ by $\frac{2}{2}$ .

$f (\frac{5}{2}) = \frac{25}{2} + \frac{- 25}{1} \cdot \frac{2}{2} + 6$

Step 3.2.2.3

Multiply $\frac{- 25}{1}$ by $\frac{2}{2}$ .

$f (\frac{5}{2}) = \frac{25}{2} + \frac{- 25 \cdot 2}{2} + 6$

Step 3.2.2.4

Write $6$ as a fraction with denominator $1$ .

$f (\frac{5}{2}) = \frac{25}{2} + \frac{- 25 \cdot 2}{2} + \frac{6}{1}$

Step 3.2.2.5

Multiply $\frac{6}{1}$ by $\frac{2}{2}$ .

$f (\frac{5}{2}) = \frac{25}{2} + \frac{- 25 \cdot 2}{2} + \frac{6}{1} \cdot \frac{2}{2}$

Step 3.2.2.6

Multiply $\frac{6}{1}$ by $\frac{2}{2}$ .

$f (\frac{5}{2}) = \frac{25}{2} + \frac{- 25 \cdot 2}{2} + \frac{6 \cdot 2}{2}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (\frac{5}{2}) = \frac{25 - 25 \cdot 2 + 6 \cdot 2}{2}$

Step 3.2.4

Simplify each term.

Tap for more steps...

Step 3.2.4.1

Multiply $- 25$ by $2$ .

$f (\frac{5}{2}) = \frac{25 - 50 + 6 \cdot 2}{2}$

Step 3.2.4.2

Multiply $6$ by $2$ .

$f (\frac{5}{2}) = \frac{25 - 50 + 12}{2}$

Step 3.2.5

Simplify the expression.

Tap for more steps...

Step 3.2.5.1

Subtract $50$ from $25$ .

$f (\frac{5}{2}) = \frac{- 25 + 12}{2}$

Step 3.2.5.2

Add $- 25$ and $12$ .

$f (\frac{5}{2}) = \frac{- 13}{2}$

Step 3.2.5.3

Move the negative in front of the fraction.

$f (\frac{5}{2}) = - \frac{13}{2}$

Step 3.2.6

The final answer is $- \frac{13}{2}$ .

$- \frac{13}{2}$

Step 4

Use the $x$ and $y$ values to find where the minimum occurs.

$(\frac{5}{2}, - \frac{13}{2})$

Step 5