Algebra Examples

$f (x) = 2 x^{2} + 6$

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

$y = 2 x^{2} + 6$

Complete the square on the right side of the equation.

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

$a = 2, b = 0, c = 6$

Consider the vertex form of a parabola.

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

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Multiply $2$ by $2$ .

$d = \frac{0}{2 \cdot 2}$

Reduce the expression by cancelling the common factors.

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Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot 2}$

Cancel the common factors.

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

$d = \frac{2 (0)}{2 (2)}$

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot 2}$

Rewrite the expression.

$d = \frac{0}{2}$

Divide $0$ by $2$ .

$d = 0$

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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Multiply $4$ by $2$ .

$e = 6 - \frac{{(0)}^{2}}{4 \cdot 2}$

Simplify the numerator.

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Remove parentheses around $0$ .

$e = 6 - \frac{0^{2}}{4 \cdot 2}$

Raising $0$ to any positive power yields $0$ .

$e = 6 - \frac{0}{4 \cdot 2}$

Simplify the denominator.

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Remove parentheses.

$e = 6 - \frac{0}{4 (2)}$

Multiply $4$ by $2$ .

$e = 6 - \frac{0}{4 \cdot 2}$

Multiply $4$ by $2$ .

$e = 6 - \frac{0}{8}$

Divide $0$ by $8$ .

$e = 6 - 0$

Multiply $- 1$ by $0$ .

$e = 6 + 0$

Add $6$ and $0$ .

$e = 6$

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

$2 {(x + 0)}^{2} + 6$

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