Find the Vertex Form f(x)=4x^2+2x-5

$f (x) = 4 x^{2} + 2 x - 5$

Step 1

Write $f (x) = 4 x^{2} + 2 x - 5$ as an equation.

$y = 4 x^{2} + 2 x - 5$

Step 2

Complete the square for $4 x^{2} + 2 x - 5$ .

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

$a = 4$

$b = 2$

$c = - 5$

Consider the vertex form of a parabola.

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

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

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

$d = \frac{2}{2 \cdot 4}$

Cancel the common factor of $2$ .

Cancel the common factor.

$d = \frac{2}{2 \cdot 4}$

Rewrite the expression.

$d = \frac{1}{4}$

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

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

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

Simplify the right side.

Simplify each term.

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

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

Multiply $4$ by $4$ .

$e = - 5 - \frac{4}{16}$

Cancel the common factor of $4$ and $16$ .

Factor $4$ out of $4$ .

$e = - 5 - \frac{4 (1)}{16}$

Cancel the common factors.

Factor $4$ out of $16$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Combine $- 5$ and $\frac{4}{4}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 5$ by $4$ .

$e = \frac{- 20 - 1}{4}$

Subtract $1$ from $- 20$ .

$e = \frac{- 21}{4}$

Move the negative in front of the fraction.

$e = - \frac{21}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x + \frac{1}{4})}^{2} - \frac{21}{4}$ .

$4 {(x + \frac{1}{4})}^{2} - \frac{21}{4}$

Step 3

Set $y$ equal to the new right side.

$y = 4 {(x + \frac{1}{4})}^{2} - \frac{21}{4}$

Find the Vertex Form f(x)=4x^2+2x-5

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