Question

What is the discriminant of `2x^2-7x-4=0` and what does that mean?

Answer 1

The discriminant of `2x^2-7x-4=0` is `81` and this means that there are 2 Real solutions for `x` to this equation.

Explanation:

The discriminant for a quadratic equation in the form
`color(white)("XXXX")` `ax^2+bx+c = 0`
is
`color(white)("XXXX")` `Delta = b^2-4ac`

`Delta { (<0, "no Real solutions"), (=0, "exactly 1 Real solution"), (>0, "2 Real solutions") :}`

For the given equation: `2x^2-7x-4 =0`

`Delta = (-7)^2 - 4(2)(-4)`
`color(white)("XXXX")` `= 49+32`
`color(white)("XXXX")` `= 81`
which tells us that there are 2 Real solutions

Answer 2

Solve `y = 2x^2 - 7x - 4 = 0`

Explanation:

`D = d^2 = b^2 - 4ac = 49 + 32 = 81` --> `d = +- 9`

This mean there are 2 real roots (2 x-intercepts). They are given by the formula:
`x = -b/(2a) +- d/(2a)`
`x = 7/ 4 +- 9/4`
`x1 = 16/4 = 4`
`x2 = -2/4 = - 1/2`