Algebra Examples

$- \frac{1}{2} {(x - 1)}^{2} + 3$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$f (x) = - \frac{1}{2} \cdot ((x - 1) (x - 1)) + 3$

Step 1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$f (x) = - \frac{1}{2} \cdot (x (x - 1) - 1 (x - 1)) + 3$

Step 1.2.2

Apply the distributive property.

$f (x) = - \frac{1}{2} \cdot (x \cdot x + x \cdot - 1 - 1 (x - 1)) + 3$

Step 1.2.3

Apply the distributive property.

$f (x) = - \frac{1}{2} \cdot (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) + 3$

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 $x$ .

$f (x) = - \frac{1}{2} \cdot (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) + 3$

Step 1.3.1.2

Move $- 1$ to the left of $x$ .

$f (x) = - \frac{1}{2} \cdot (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) + 3$

Step 1.3.1.3

Rewrite $- 1 x$ as $- x$ .

$f (x) = - \frac{1}{2} \cdot (x^{2} - x - 1 x - 1 \cdot - 1) + 3$

Step 1.3.1.4

Rewrite $- 1 x$ as $- x$ .

$f (x) = - \frac{1}{2} \cdot (x^{2} - x - x - 1 \cdot - 1) + 3$

Step 1.3.1.5

Multiply $- 1$ by $- 1$ .

$f (x) = - \frac{1}{2} \cdot (x^{2} - x - x + 1) + 3$

Step 1.3.2

Subtract $x$ from $- x$ .

$f (x) = - \frac{1}{2} \cdot (x^{2} - 2 x + 1) + 3$

Step 1.4

Apply the distributive property.

$f (x) = - \frac{1}{2} x^{2} - \frac{1}{2} \cdot (- 2 x) - \frac{1}{2} \cdot 1 + 3$

Step 1.5

Simplify.

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Step 1.5.1

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

$f (x) = - \frac{x^{2}}{2} - \frac{1}{2} \cdot (- 2 x) - \frac{1}{2} \cdot 1 + 3$

Step 1.5.2

Cancel the common factor of $2$ .

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Step 1.5.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$f (x) = - \frac{x^{2}}{2} + \frac{- 1}{2} \cdot (- 2 x) - \frac{1}{2} \cdot 1 + 3$

Step 1.5.2.2

Factor $2$ out of $- 2 x$ .

$f (x) = - \frac{x^{2}}{2} + \frac{- 1}{2} \cdot (2 (- x)) - \frac{1}{2} \cdot 1 + 3$

Step 1.5.2.3

Cancel the common factor.

$f (x) = - \frac{x^{2}}{2} + \frac{- 1}{2} \cdot (2 (- x)) - \frac{1}{2} \cdot 1 + 3$

Step 1.5.2.4

Rewrite the expression.

$f (x) = - \frac{x^{2}}{2} - 1 (- x) - \frac{1}{2} \cdot 1 + 3$

Step 1.5.3

Multiply $- 1$ by $- 1$ .

$f (x) = - \frac{x^{2}}{2} + 1 x - \frac{1}{2} \cdot 1 + 3$

Step 1.5.4

Multiply $x$ by $1$ .

$f (x) = - \frac{x^{2}}{2} + x - \frac{1}{2} \cdot 1 + 3$

Step 1.5.5

Multiply $- 1$ by $1$ .

$f (x) = - \frac{x^{2}}{2} + x - \frac{1}{2} + 3$

Step 2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (x) = - \frac{x^{2}}{2} + x - \frac{1}{2} + 3 \cdot \frac{2}{2}$

Step 3

Combine $3$ and $\frac{2}{2}$ .

$f (x) = - \frac{x^{2}}{2} + x - \frac{1}{2} + \frac{3 \cdot 2}{2}$

Step 4

Combine the numerators over the common denominator.

$f (x) = - \frac{x^{2}}{2} + x + \frac{- 1 + 3 \cdot 2}{2}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $3$ by $2$ .

$f (x) = - \frac{x^{2}}{2} + x + \frac{- 1 + 6}{2}$

Step 5.2

Add $- 1$ and $6$ .

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

Step 6

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

$f_{max}$ $x = a x^{2} + b x + c$ 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 $b$ .

$x = - \frac{1}{2 (- 0.5)}$

Step 7.2

Remove parentheses.

$x = - \frac{1}{2 (- 0.5)}$

Step 7.3

Simplify $- \frac{1}{2 (- 0.5)}$ .

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Step 7.3.1

Multiply $2$ by $- 0.5$ .

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

Step 7.3.2

Divide $1$ by $- 1$ .

$x = - - 1$

Step 7.3.3

Multiply $- 1$ by $- 1$ .

$x = 1$

Step 8

Evaluate $f (1)$ .

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Step 8.1

Replace the variable $x$ with $1$ in the expression.

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

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

Remove parentheses.

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

Step 8.2.1.2

Combine the numerators over the common denominator.

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

Step 8.2.2

Simplify each term.

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Step 8.2.2.1

One to any power is one.

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

Step 8.2.2.2

Multiply $- 1$ by $1$ .

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

Step 8.2.3

Simplify the expression.

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Step 8.2.3.1

Add $- 1$ and $5$ .

$f (1) = 1 + \frac{4}{2}$

Step 8.2.3.2

Divide $4$ by $2$ .

$f (1) = 1 + 2$

Step 8.2.3.3

Add $1$ and $2$ .

$f (1) = 3$

Step 8.2.4

The final answer is $3$ .

$3$

Step 9

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

$(1, 3)$

Step 10