Algebra Examples

$f (x) = 5 x^{2} + 2 x - 5$

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

$y = 5 x^{2} + 2 x - 5$

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 $c$ .

$a = 5, b = 2, c = - 5$

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 $5$ to get $2 \cdot 5$ .

$d = \frac{2}{2 \cdot 5}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$d = \frac{2}{2 \cdot 5}$

Rewrite the expression.

$d = \frac{1}{5}$

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 $5$ to get $4 \cdot 5$ .

$e = - 5 - \frac{{(2)}^{2}}{4 \cdot 5}$

Simplify the numerator.

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Remove parentheses around $2$ .

$e = - 5 - \frac{2^{2}}{4 \cdot 5}$

Raise $2$ to the power of $2$ to get $4$ .

$e = - 5 - \frac{4}{4 \cdot 5}$

Simplify the denominator.

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Remove parentheses.

$e = - 5 - \frac{4}{4 (5)}$

Multiply $4$ by $5$ to get $4 \cdot 5$ .

$e = - 5 - \frac{4}{4 \cdot 5}$

Multiply $4$ by $5$ to get $20$ .

$e = - 5 - \frac{4}{20}$

Reduce the expression by cancelling the common factors.

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Factor $4$ out of $20$ .

$e = - 5 - \frac{4 \cdot 1}{4 \cdot 5}$

Cancel the common factor.

$e = - 5 - \frac{4 \cdot 1}{4 \cdot 5}$

Rewrite the expression.

$e = - 5 - \frac{1}{5}$

To write $\frac{- 5}{1}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$e = \frac{- 5}{1} \cdot \frac{5}{5} - \frac{1}{5}$

Write each expression with a common denominator of $5$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$e = \frac{- 5 \cdot 5}{1 \cdot 5} - \frac{1}{5}$

Multiply $5$ by $1$ to get $5$ .

$e = \frac{- 5 \cdot 5}{5} - \frac{1}{5}$

Combine the numerators over the common denominator.

$e = \frac{- 5 \cdot 5 - 1 \cdot 1}{5}$

Simplify the numerator.

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Multiply $- 5$ by $5$ to get $- 25$ .

$e = \frac{- 25 - 1 \cdot 1}{5}$

Multiply $- 1$ by $1$ to get $- 1$ .

$e = \frac{- 25 - 1}{5}$

Subtract $1$ from $- 25$ to get $- 26$ .

$e = \frac{- 26}{5}$

Move the negative in front of the fraction.

$e = - \frac{26}{5}$

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

$5 {(x + \frac{1}{5})}^{2} - \frac{26}{5}$

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