Question

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

Answer 1

The vertex is `(-7/5,-79/5)` `=` `(-1.4,-15.8)`

Explanation:

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

`y=ax^2+bx+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=(-b)/(2a)`

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

`x=(-14)/(2*5)`

Simplify.

`x=(-14)/(10)`

Reduce.

`x=-7/5=-1.4`

To find the `y`-value of the vertex, subsitute `-7/5` for `x` and solve for `y`.

`y=5(-7/5)^2+14(-7/5)-6`

Simplify.

`y=5(49/25)-98/5-6`

Simplify.

`y=245/25-98/5-6`

Reduce `245/25` by dividing the numerator and denominator by `5`.

`y=((245-:5)/(25-:5))-98/5-6`

Simplify.j

`y=49/5-98/5-6`

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=6/1`.

Multiply `98/5` and `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 `3/3=1`. This changes the numbers, but not the values of the fractions.

`y=49/5-98/5-6xxcolor(magenta)5/color(magenta)5`

Simplify.

`y=49/5-98/5-30/5`

Simplify.

`y=(49-98-30)/5`

`y=-79/5=-15.8`

The vertex is `(-7/5,-79/5)` `=` `(-1.4,-15.8)`

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