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Algebra Examples

f (x) = x^{2} - 4 x + 2

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

Step 1

The minimum of a quadratic function occurs at

x = - \frac{b}{2 a}

$x = - \frac{b}{2 a}$ . If

a

$a$ is positive, the minimum value of the function is

f (- \frac{b}{2 a})

$f (- \frac{b}{2 a})$ .

f_{min}

$f_{min}$

x = a x^{2} + b x + c

$x = a x^{2} + b x + c$ occurs at

x = - \frac{b}{2 a}

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

Step 2

Find the value of

x = - \frac{b}{2 a}

$x = - \frac{b}{2 a}$ .

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Step 2.1

Substitute in the values of

a

$a$ and

b

$b$ .

x = - \frac{- 4}{2 (1)}

$x = - \frac{- 4}{2 (1)}$

Step 2.2

Remove parentheses.

x = - \frac{- 4}{2 (1)}

$x = - \frac{- 4}{2 (1)}$

Step 2.3

Simplify

- \frac{- 4}{2 (1)}

$- \frac{- 4}{2 (1)}$ .

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Step 2.3.1

Cancel the common factor of

- 4

$- 4$ and

2

$2$ .

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Step 2.3.1.1

Factor

2

$2$ out of

- 4

$- 4$ .

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

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

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor

2

$2$ out of

2 \cdot 1

$2 \cdot 1$ .

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

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

Step 2.3.1.2.2

Cancel the common factor.

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

Step 2.3.1.2.3

Rewrite the expression.

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

Step 2.3.1.2.4

Divide

$- 2$ by

$1$ .

$x = - - 2$

$x = - - 2$

$x = - - 2$

Step 2.3.2

Multiply

$- 1$ by

$- 2$ .

$x = 2$

$x = 2$

$x = 2$

Step 3

Evaluate

$f (2)$ .

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Step 3.1

Replace the variable

$x$ with

$2$ in the expression.

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

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

Raise

$2$ to the power of

$2$ .

$f (2) = 4 - 4 \cdot 2 + 2$

Step 3.2.1.2

Multiply

$- 4$ by

$2$ .

$f (2) = 4 - 8 + 2$

$f (2) = 4 - 8 + 2$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract

$8$ from

$4$ .

$f (2) = - 4 + 2$

Step 3.2.2.2

Add

$- 4$ and

$2$ .

$f (2) = - 2$

$f (2) = - 2$

Step 3.2.3

The final answer is

$- 2$ .

$- 2$

$- 2$

$- 2$

Step 4

Use the

$x$ and

$y$ values to find where the minimum occurs.

$(2, - 2)$

Step 5

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$