What is the vertex form of y=2x^2+7x+3 ?

Aug 12, 2017

The vertex form is $y = 2 {\left(x + \frac{7}{4}\right)}^{2} - \frac{25}{8}$.

Explanation:

$y = 2 {x}^{2} + 7 x + 3$ is a quadratic equation in standard form:

$y = a {x}^{2} + b x + c$, where $a = 2$, $b = 7$, and $c = 3$.

The vertex form is $y = a {\left(x - h\right)}^{2} + k$, where $\left(h , k\right)$ is the vertex.

In order to determine $h$ from the standard form, use this formula:

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

$h = x = \frac{- 7}{2 \cdot 2}$

$h = x = - \frac{7}{4}$

To determine $k$, substitute the value of $h$ for $x$ and solve. $f \left(h\right) = y = k$

Substitute $- \frac{7}{4}$ for $x$ and solve.

$k = 2 {\left(- \frac{7}{4}\right)}^{2} + 7 \left(- \frac{7}{4}\right) + 3$

$k = 2 \left(\frac{49}{16}\right) - \frac{49}{4} + 3$

$k = \frac{98}{16} - \frac{49}{4} + 3$

Divide $\frac{98}{16}$ by color(teal)(2/2

$k = \frac{98 \div \textcolor{t e a l}{2}}{16 \div \textcolor{t e a l}{2}} - \frac{49}{4} + 3$

Simplify.

$k = \frac{49}{8} - \frac{49}{4} + 3$

The least common denominator is $8$. Multiply $\frac{49}{4}$ and $3$ by equivalent fractions to give them a denominator of $8$.

k=49/8-49/4xxcolor(red)(2/2)+3xxcolor(blue)(8/8

$k = \frac{49}{8} - \frac{98}{8} + \frac{24}{8}$

$k = - \frac{25}{8}$

The vertex form of the quadratic equation is:

$y = 2 {\left(x + \frac{7}{4}\right)}^{2} - \frac{25}{8}$

graph{y=2x^2+7x+3 [-10, 10, -5, 5]}