Question

How do you find the discriminant for `x^2-4/5x=3` and determine the number and type of solutions?

Answer 1

See a solution process below:

Explanation:

First, rewrite the equation in standard form as:

`x^2 - 4/5x - 3 = 0`

The quadratic formula states:

For `ax^2 + bx + c = 0`, the values of `x` which are the solutions to the equation are given by:

`x = (-b +- sqrt(b^2 - 4ac))/(2a)`

The discriminate is the portion of the quadratic equation within the radical: `color(blue)(b)^2 - 4color(red)(a)color(green)(c)`

If the discriminate is:
- Positive, you will get two real solutions
- Zero you get just ONE solution
- Negative you get complex solutions

To find the discriminant for this problem substitute:

`color(red)(1)` for `color(red)(a)`

`color(blue)(-4/5)` for `color(blue)(b)`

`color(green)(3)` for `color(green)(c)`

`color(blue)(-3)^2 - (4 * color(red)(1) * color(green)(-4/5))`

`9 - (-16/5)`

`9 + 16/5`

`9 + 15/5 + 1/5`

`9 + 3 + 1/5`

`12 + 1/5`

`12 1/5` or `61/5`

Because the discriminate is positive there will two (2) real solutions for this equation.