Algebra Examples

$f (x) = 0.25 {(2 x - 15)}^{2} + 150$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(2 x - 15)}^{2}$ as $(2 x - 15) (2 x - 15)$ .

$f (x) = 0.25 ((2 x - 15) (2 x - 15)) + 150$

Step 1.2

Expand $(2 x - 15) (2 x - 15)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$f (x) = 0.25 (2 x (2 x - 15) - 15 (2 x - 15)) + 150$

Step 1.2.2

Apply the distributive property.

$f (x) = 0.25 (2 x (2 x) + 2 x \cdot - 15 - 15 (2 x - 15)) + 150$

Step 1.2.3

Apply the distributive property.

$f (x) = 0.25 (2 x (2 x) + 2 x \cdot - 15 - 15 (2 x) - 15 \cdot - 15) + 150$

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

Rewrite using the commutative property of multiplication.

$f (x) = 0.25 (2 \cdot (2 x \cdot x) + 2 x \cdot - 15 - 15 (2 x) - 15 \cdot - 15) + 150$

Step 1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.1.2.1

Move $x$ .

$f (x) = 0.25 (2 \cdot (2 (x \cdot x)) + 2 x \cdot - 15 - 15 (2 x) - 15 \cdot - 15) + 150$

Step 1.3.1.2.2

Multiply $x$ by $x$ .

$f (x) = 0.25 (2 \cdot (2 x^{2}) + 2 x \cdot - 15 - 15 (2 x) - 15 \cdot - 15) + 150$

Step 1.3.1.3

Multiply $2$ by $2$ .

$f (x) = 0.25 (4 x^{2} + 2 x \cdot - 15 - 15 (2 x) - 15 \cdot - 15) + 150$

Step 1.3.1.4

Multiply $- 15$ by $2$ .

$f (x) = 0.25 (4 x^{2} - 30 x - 15 (2 x) - 15 \cdot - 15) + 150$

Step 1.3.1.5

Multiply $2$ by $- 15$ .

$f (x) = 0.25 (4 x^{2} - 30 x - 30 x - 15 \cdot - 15) + 150$

Step 1.3.1.6

Multiply $- 15$ by $- 15$ .

$f (x) = 0.25 (4 x^{2} - 30 x - 30 x + 225) + 150$

Step 1.3.2

Subtract $30 x$ from $- 30 x$ .

$f (x) = 0.25 (4 x^{2} - 60 x + 225) + 150$

Step 1.4

Apply the distributive property.

$f (x) = 0.25 (4 x^{2}) + 0.25 (- 60 x) + 0.25 \cdot 225 + 150$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $0.25 (4 x^{2})$ .

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Step 1.5.1.1

Multiply $4$ by $0.25$ .

$f (x) = 1 x^{2} + 0.25 (- 60 x) + 0.25 \cdot 225 + 150$

Step 1.5.1.2

Multiply $x^{2}$ by $1$ .

$f (x) = x^{2} + 0.25 (- 60 x) + 0.25 \cdot 225 + 150$

Step 1.5.2

Multiply $- 60$ by $0.25$ .

$f (x) = x^{2} - 15 x + 0.25 \cdot 225 + 150$

Step 1.5.3

Multiply $0.25$ by $225$ .

$f (x) = x^{2} - 15 x + 56.25 + 150$

Step 2

Add $56.25$ and $150$ .

$f (x) = x^{2} - 15 x + 206.25$

Step 3

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 4

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

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Step 4.1

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

$x = - \frac{- 15}{2 (1)}$

Step 4.2

Remove parentheses.

$x = - \frac{- 15}{2 (1)}$

Step 4.3

Simplify $- \frac{- 15}{2 (1)}$ .

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Step 4.3.1

Multiply $2$ by $1$ .

$x = - \frac{- 15}{2}$

Step 4.3.2

Move the negative in front of the fraction.

$x = - - \frac{15}{2}$

Step 4.3.3

Multiply $- - \frac{15}{2}$ .

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Step 4.3.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{15}{2})$

Step 4.3.3.2

Multiply $\frac{15}{2}$ by $1$ .

$x = \frac{15}{2}$

Step 5

Evaluate $f (\frac{15}{2})$ .

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Step 5.1

Replace the variable $x$ with $\frac{15}{2}$ in the expression.

$f (\frac{15}{2}) = {(\frac{15}{2})}^{2} - 15 (\frac{15}{2}) + 206.25$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Apply the product rule to $\frac{15}{2}$ .

$f (\frac{15}{2}) = \frac{15^{2}}{2^{2}} - 15 (\frac{15}{2}) + 206.25$

Step 5.2.1.2

Raise $15$ to the power of $2$ .

$f (\frac{15}{2}) = \frac{225}{2^{2}} - 15 (\frac{15}{2}) + 206.25$

Step 5.2.1.3

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

$f (\frac{15}{2}) = \frac{225}{4} - 15 (\frac{15}{2}) + 206.25$

Step 5.2.1.4

Multiply $- 15 (\frac{15}{2})$ .

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Step 5.2.1.4.1

Combine $- 15$ and $\frac{15}{2}$ .

$f (\frac{15}{2}) = \frac{225}{4} + \frac{- 15 \cdot 15}{2} + 206.25$

Step 5.2.1.4.2

Multiply $- 15$ by $15$ .

$f (\frac{15}{2}) = \frac{225}{4} + \frac{- 225}{2} + 206.25$

Step 5.2.1.5

Cancel the common factor of $- 225$ and $2$ .

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Step 5.2.1.5.1

Rewrite $- 225$ as $1 (- 225)$ .

$f (\frac{15}{2}) = \frac{225}{4} + \frac{1 (- 225)}{2} + 206.25$

Step 5.2.1.5.2

Cancel the common factors.

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Step 5.2.1.5.2.1

Rewrite $2$ as $1 (2)$ .

$f (\frac{15}{2}) = \frac{225}{4} + \frac{1 \cdot - 225}{1 \cdot 2} + 206.25$

Step 5.2.1.5.2.2

Cancel the common factor.

$f (\frac{15}{2}) = \frac{225}{4} + \frac{1 \cdot - 225}{1 \cdot 2} + 206.25$

Step 5.2.1.5.2.3

Rewrite the expression.

$f (\frac{15}{2}) = \frac{225}{4} + \frac{- 225}{2} + 206.25$

Step 5.2.1.6

Move the negative in front of the fraction.

$f (\frac{15}{2}) = \frac{225}{4} - \frac{225}{2} + 206.25$

Step 5.2.2

Find the common denominator.

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Step 5.2.2.1

Multiply $\frac{225}{2}$ by $\frac{2}{2}$ .

$f (\frac{15}{2}) = \frac{225}{4} - (\frac{225}{2} \cdot \frac{2}{2}) + 206.25$

Step 5.2.2.2

Multiply $\frac{225}{2}$ by $\frac{2}{2}$ .

$f (\frac{15}{2}) = \frac{225}{4} - \frac{225 \cdot 2}{2 \cdot 2} + 206.25$

Step 5.2.2.3

Write $206.25$ as a fraction with denominator $1$ .

$f (\frac{15}{2}) = \frac{225}{4} - \frac{225 \cdot 2}{2 \cdot 2} + \frac{206.25}{1}$

Step 5.2.2.4

Multiply $\frac{206.25}{1}$ by $\frac{4}{4}$ .

$f (\frac{15}{2}) = \frac{225}{4} - \frac{225 \cdot 2}{2 \cdot 2} + \frac{206.25}{1} \cdot \frac{4}{4}$

Step 5.2.2.5

Multiply $\frac{206.25}{1}$ by $\frac{4}{4}$ .

$f (\frac{15}{2}) = \frac{225}{4} - \frac{225 \cdot 2}{2 \cdot 2} + \frac{206.25 \cdot 4}{4}$

Step 5.2.2.6

Multiply $2$ by $2$ .

$f (\frac{15}{2}) = \frac{225}{4} - \frac{225 \cdot 2}{4} + \frac{206.25 \cdot 4}{4}$

Step 5.2.3

Combine the numerators over the common denominator.

$f (\frac{15}{2}) = \frac{225 - 225 \cdot 2 + 206.25 \cdot 4}{4}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Multiply $- 225$ by $2$ .

$f (\frac{15}{2}) = \frac{225 - 450 + 206.25 \cdot 4}{4}$

Step 5.2.4.2

Multiply $206.25$ by $4$ .

$f (\frac{15}{2}) = \frac{225 - 450 + 825}{4}$

Step 5.2.5

Simplify the expression.

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Step 5.2.5.1

Subtract $450$ from $225$ .

$f (\frac{15}{2}) = \frac{- 225 + 825}{4}$

Step 5.2.5.2

Add $- 225$ and $825$ .

$f (\frac{15}{2}) = \frac{600}{4}$

Step 5.2.5.3

Divide $600$ by $4$ .

$f (\frac{15}{2}) = 150$

Step 5.2.6

The final answer is $150$ .

$150$

Step 6

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

$(\frac{15}{2}, 150)$

Step 7