Algebra Examples

$x^{2} + 10 x + 10$

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 $x = - \frac{b}{2 a}$ .

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Step 2.1

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

$x = - \frac{10}{2 (1)}$

Step 2.2

Remove parentheses.

$x = - \frac{10}{2 (1)}$

Step 2.3

Simplify $- \frac{10}{2 (1)}$ .

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Step 2.3.1

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

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Step 2.3.1.1

Factor $2$ out of $10$ .

$x = - \frac{2 \cdot 5}{2 \cdot 1}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $2 \cdot 1$ .

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

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot 5}{2 \cdot 1}$

Step 2.3.1.2.3

Rewrite the expression.

$x = - \frac{5}{1}$

Step 2.3.1.2.4

Divide $5$ by $1$ .

$x = - 1 \cdot 5$

Step 2.3.2

Multiply $- 1$ by $5$ .

$x = - 5$

Step 3

Evaluate $f (- 5)$ .

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Step 3.1

Replace the variable $x$ with $- 5$ in the expression.

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

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

$f (- 5) = 25 + 10 (- 5) + 10$

Step 3.2.1.2

Multiply $10$ by $- 5$ .

$f (- 5) = 25 - 50 + 10$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $50$ from $25$ .

$f (- 5) = - 25 + 10$

Step 3.2.2.2

Add $- 25$ and $10$ .

$f (- 5) = - 15$

Step 3.2.3

The final answer is $- 15$ .

$- 15$

Step 4

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

$(- 5, - 15)$

Step 5