# What is the the vertex of y =5x^2+14x-6 ?

Mar 7, 2018

The vertex is $\left(- \frac{7}{5} , - \frac{79}{5}\right)$$=$$\left(- 1.4 , - 15.8\right)$

#### Explanation:

$y = 5 {x}^{2} + 14 x - 6$ is a quadratic equation in standard form:

$y = a {x}^{2} + b x + c ,$

where:

$a = 5 ,$ $b = 14 ,$ $c = - 6$

The vertex is the minimum or maximum point on a parabola. To find the vertex of a quadratic equation in standard form, determine the axis of symmetry, which will be the $x$-value of the vertex.

Axis of symmetry: vertical line that divides the parabola into two equal halves. The formula for the axis of symmetry for a quadratic equation in standard form is:

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

Plug in the known values and solve for $x$.

$x = \frac{- 14}{2 \cdot 5}$

Simplify.

$x = \frac{- 14}{10}$

Reduce.

$x = - \frac{7}{5} = - 1.4$

To find the $y$-value of the vertex, subsitute $- \frac{7}{5}$ for $x$ and solve for $y$.

$y = 5 {\left(- \frac{7}{5}\right)}^{2} + 14 \left(- \frac{7}{5}\right) - 6$

Simplify.

$y = 5 \left(\frac{49}{25}\right) - \frac{98}{5} - 6$

Simplify.

$y = \frac{245}{25} - \frac{98}{5} - 6$

Reduce $\frac{245}{25}$ by dividing the numerator and denominator by $5$.

$y = \left(\frac{245 \div 5}{25 \div 5}\right) - \frac{98}{5} - 6$

Simplify.j

$y = \frac{49}{5} - \frac{98}{5} - 6$

In order to add or subtract fractions, they must have a common denominator, called the least common denominator (LCD). In this case, the LCD is $5$. Recall that a whole number has a denominator of $1$, so $6 = \frac{6}{1}$.

Multiply $\frac{98}{5}$ and $\frac{6}{1}$ by a fractional form of $1$ that will give them the LCD of $5$. An example of a fractional form of $1$ is $\frac{3}{3} = 1$. This changes the numbers, but not the values of the fractions.

$y = \frac{49}{5} - \frac{98}{5} - 6 \times \frac{\textcolor{m a \ge n t a}{5}}{\textcolor{m a \ge n t a}{5}}$

Simplify.

$y = \frac{49}{5} - \frac{98}{5} - \frac{30}{5}$

Simplify.

$y = \frac{49 - 98 - 30}{5}$

$y = - \frac{79}{5} = - 15.8$

The vertex is $\left(- \frac{7}{5} , - \frac{79}{5}\right)$$=$$\left(- 1.4 , - 15.8\right)$

graph{y=5x^2+14x-6 [-14.36, 14.11, -20.68, -6.44]}