Algebra Examples

$f (x) = \frac{1}{2} {(x + 5)}^{2} - 9$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$f (x) = \frac{1}{2} \cdot ((x + 5) (x + 5)) - 9$

Step 1.2

Expand $(x + 5) (x + 5)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$f (x) = \frac{1}{2} \cdot (x (x + 5) + 5 (x + 5)) - 9$

Step 1.2.2

Apply the distributive property.

$f (x) = \frac{1}{2} \cdot (x \cdot x + x \cdot 5 + 5 (x + 5)) - 9$

Step 1.2.3

Apply the distributive property.

$f (x) = \frac{1}{2} \cdot (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) - 9$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$f (x) = \frac{1}{2} \cdot (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) - 9$

Step 1.3.1.2

Move $5$ to the left of $x$ .

$f (x) = \frac{1}{2} \cdot (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5) - 9$

Step 1.3.1.3

Multiply $5$ by $5$ .

$f (x) = \frac{1}{2} \cdot (x^{2} + 5 x + 5 x + 25) - 9$

Step 1.3.2

Add $5 x$ and $5 x$ .

$f (x) = \frac{1}{2} \cdot (x^{2} + 10 x + 25) - 9$

Step 1.4

Apply the distributive property.

$f (x) = \frac{1}{2} x^{2} + \frac{1}{2} \cdot (10 x) + \frac{1}{2} \cdot 25 - 9$

Step 1.5

Simplify.

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Step 1.5.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$f (x) = \frac{x^{2}}{2} + \frac{1}{2} \cdot (10 x) + \frac{1}{2} \cdot 25 - 9$

Step 1.5.2

Cancel the common factor of $2$ .

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Step 1.5.2.1

Factor $2$ out of $10 x$ .

$f (x) = \frac{x^{2}}{2} + \frac{1}{2} \cdot (2 (5 x)) + \frac{1}{2} \cdot 25 - 9$

Step 1.5.2.2

Cancel the common factor.

$f (x) = \frac{x^{2}}{2} + \frac{1}{2} \cdot (2 (5 x)) + \frac{1}{2} \cdot 25 - 9$

Step 1.5.2.3

Rewrite the expression.

$f (x) = \frac{x^{2}}{2} + 5 x + \frac{1}{2} \cdot 25 - 9$

Step 1.5.3

Combine $\frac{1}{2}$ and $25$ .

$f (x) = \frac{x^{2}}{2} + 5 x + \frac{25}{2} - 9$

Step 2

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (x) = \frac{x^{2}}{2} + 5 x + \frac{25}{2} - 9 \cdot \frac{2}{2}$

Step 3

Combine $- 9$ and $\frac{2}{2}$ .

$f (x) = \frac{x^{2}}{2} + 5 x + \frac{25}{2} + \frac{- 9 \cdot 2}{2}$

Step 4

Combine the numerators over the common denominator.

$f (x) = \frac{x^{2}}{2} + 5 x + \frac{25 - 9 \cdot 2}{2}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 9$ by $2$ .

$f (x) = \frac{x^{2}}{2} + 5 x + \frac{25 - 18}{2}$

Step 5.2

Subtract $18$ from $25$ .

$f (x) = \frac{x^{2}}{2} + 5 x + \frac{7}{2}$

Step 6

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 7

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

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Step 7.1

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

$x = - \frac{5}{2 (0.5)}$

Step 7.2

Remove parentheses.

$x = - \frac{5}{2 (0.5)}$

Step 7.3

Simplify $- \frac{5}{2 (0.5)}$ .

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Step 7.3.1

Multiply $2$ by $0.5$ .

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

Step 7.3.2

Divide $5$ by $1$ .

$x = - 1 \cdot 5$

Step 7.3.3

Multiply $- 1$ by $5$ .

$x = - 5$

Step 8

Evaluate $f (- 5)$ .

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Step 8.1

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

$f (- 5) = \frac{{(- 5)}^{2}}{2} + 5 (- 5) + \frac{7}{2}$

Step 8.2

Simplify the result.

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Step 8.2.1

Combine fractions.

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Step 8.2.1.1

Combine the numerators over the common denominator.

$f (- 5) = 5 (- 5) + \frac{{(- 5)}^{2} + 7}{2}$

Step 8.2.1.2

Simplify the expression.

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Step 8.2.1.2.1

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

$f (- 5) = 5 (- 5) + \frac{25 + 7}{2}$

Step 8.2.1.2.2

Add $25$ and $7$ .

$f (- 5) = 5 (- 5) + \frac{32}{2}$

Step 8.2.2

Simplify each term.

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Step 8.2.2.1

Multiply $5$ by $- 5$ .

$f (- 5) = - 25 + \frac{32}{2}$

Step 8.2.2.2

Divide $32$ by $2$ .

$f (- 5) = - 25 + 16$

Step 8.2.3

Add $- 25$ and $16$ .

$f (- 5) = - 9$

Step 8.2.4

The final answer is $- 9$ .

$- 9$

Step 9

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

$(- 5, - 9)$

Step 10