Question

What is the discriminant of `-9x^2+10x=-2x+4` and what does that mean?

Answer 1

`0`
It means that there is exactly 1 Real solution for this equation

Explanation:

The discriminant of a quadratic equation is `b^2 – 4ac`. To calculate the discriminant of the equation you provided, we move `-2x` and `4` to the left, resulting in `-9x^2+12x-4`. To calculate the discriminant of this simplified equation, we use our formula above, but substitute `12` for `b`, `-9` as `a`, and `-4` as `c`.

We get this equation: `(12)^2 - 4(-9)(-4)`, which evaluates to `0`

The "meaning" is the result of the discriminant being a component of the quadratic formula for the solution(s) to quadratic equation in the form:
`color(white)("XXXX")` `ax^2+bx+c=0`
where the solutions can be determined by:
`color(white)("XXXX")` `x=(-b+-sqrt(b^2-4ac))/(2a)`

Notice that the discriminant is the component within the square root, and as a result:
`"discriminant" { (= 0, " one Real root"), (< 0, " no Real Roots"), (> 0, " two Real roots") :}`