Algebra Examples

$g (x) = 2 x^{2} - 4 x + 4$

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 $b$ .

$x = - \frac{- 4}{2 (2)}$

Step 2.2

Remove parentheses.

$x = - \frac{- 4}{2 (2)}$

Step 2.3

Simplify $- \frac{- 4}{2 (2)}$ .

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Step 2.3.1

Cancel the common factor of $- 4$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $- 4$ .

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

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

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

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

Step 2.3.1.2.2

Cancel the common factor.

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

Step 2.3.1.2.3

Rewrite the expression.

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

Step 2.3.2

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

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Step 2.3.2.1

Factor $2$ out of $- 2$ .

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

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.4

Divide $- 1$ by $1$ .

$x = - - 1$

Step 2.3.3

Multiply $- 1$ by $- 1$ .

$x = 1$

Step 3

Evaluate $f (1)$ .

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Step 3.1

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

$g (1) = 2 {(1)}^{2} - 4 \cdot 1 + 4$

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

One to any power is one.

$g (1) = 2 \cdot 1 - 4 \cdot 1 + 4$

Step 3.2.1.2

Multiply $2$ by $1$ .

$g (1) = 2 - 4 \cdot 1 + 4$

Step 3.2.1.3

Multiply $- 4$ by $1$ .

$g (1) = 2 - 4 + 4$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $4$ from $2$ .

$g (1) = - 2 + 4$

Step 3.2.2.2

Add $- 2$ and $4$ .

$g (1) = 2$

Step 3.2.3

The final answer is $2$ .

$2$

Step 4

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

$(1, 2)$

Step 5