Algebra Examples

$3 - {(x - 1)}^{2}$

Step 1

Simplify each term.

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Step 1.1

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

$f (x) = 3 - ((x - 1) (x - 1))$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$f (x) = 3 - (x (x - 1) - 1 (x - 1))$

Step 1.2.2

Apply the distributive property.

$f (x) = 3 - (x \cdot x + x \cdot - 1 - 1 (x - 1))$

Step 1.2.3

Apply the distributive property.

$f (x) = 3 - (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)$

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 $x$ .

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

Step 1.3.1.2

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

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

Step 1.3.1.3

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

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

Step 1.3.1.4

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

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

Step 1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.3.2

Subtract $x$ from $- x$ .

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

Step 1.4

Apply the distributive property.

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

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 2$ by $- 1$ .

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

Step 1.5.2

Multiply $- 1$ by $1$ .

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

Step 2

Subtract $1$ from $3$ .

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

Step 3

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 4

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

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Step 4.1

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

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

Step 4.2

Remove parentheses.

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

Step 4.3

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

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Step 4.3.1

Cancel the common factor of $2$ .

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Step 4.3.1.1

Cancel the common factor.

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

Step 4.3.1.2

Rewrite the expression.

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

Step 4.3.1.3

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$x = - (- 1 \cdot 1)$

Step 4.3.2

Multiply $- (- 1 \cdot 1)$ .

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Step 4.3.2.1

Multiply $- 1$ by $1$ .

$x = - - 1$

Step 4.3.2.2

Multiply $- 1$ by $- 1$ .

$x = 1$

Step 5

Evaluate $f (1)$ .

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Step 5.1

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

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

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

One to any power is one.

$f (1) = - 1 \cdot 1 + 2 (1) + 2$

Step 5.2.1.2

Multiply $- 1$ by $1$ .

$f (1) = - 1 + 2 (1) + 2$

Step 5.2.1.3

Multiply $2$ by $1$ .

$f (1) = - 1 + 2 + 2$

Step 5.2.2

Simplify by adding numbers.

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Step 5.2.2.1

Add $- 1$ and $2$ .

$f (1) = 1 + 2$

Step 5.2.2.2

Add $1$ and $2$ .

$f (1) = 3$

Step 5.2.3

The final answer is $3$ .

$3$

Step 6

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

$(1, 3)$

Step 7