Algebra Examples

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

Replace $f (x)$ with $y$ to find the properties of the parabola.

$y = x^{2} - 4 x + 2$

Complete the square for $x^{2} - 4 x + 2$ .

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Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1, b = - 4, c = 2$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

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

Simplify the right side.

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Reduce the expression by cancelling the common factors.

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Factor $2$ out of $4$ .

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

Cancel the common factors.

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Factor $2$ out of $2 \cdot 1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

$d = - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$d = - 2$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Simplify each term.

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Reduce the expression by cancelling the common factors.

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Rewrite $- 4$ as $- 1 (4)$ .

$e = 2 - \frac{{(- 1 \cdot 4)}^{2}}{4 (1)}$

Apply the product rule to $- 1 (4)$ .

$e = 2 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 (1)}$

Raise $- 1$ to the power of $2$ .

$e = 2 - \frac{1 \cdot 4^{2}}{4 (1)}$

Multiply $4^{2}$ by $1$ .

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

Factor $4$ out of $4^{2}$ .

$e = 2 - \frac{4 \cdot 4}{4 \cdot 1}$

Cancel the common factors.

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Factor $4$ out of $4 \cdot 1$ .

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

Cancel the common factor.

$e = 2 - \frac{4 \cdot 4}{4 \cdot 1}$

Rewrite the expression.

$e = 2 - \frac{4}{1}$

Divide $4$ by $1$ .

$e = 2 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = 2 - 4$

Subtract $4$ from $2$ .

$e = - 2$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $a {(x + d)}^{2} + e$ .

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

Set $y$ equal to the new right side.

$y = {(x - 2)}^{2} - 2$

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