Question

How do you find the discriminant and how many and what type of solutions does `x^2 + 5x + 7 = 0` have?

Answer 1

See a solution process below:

Explanation:

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)(5)` for `color(blue)(b)`

`color(green)(7)` for `color(green)(c)`

Giving

`color(blue)(5)^2 - (4 * color(red)(1) * color(green)(7)) =>`

`25 - 28 =>`

`-3`

Because the number is negative you will get two complex solutions.

Answer 2

`x^2+5x+7=0` has no real roots therefore no defined solutions.

Explanation:

D = `b^2 - 4ac`

A standard polynomial form is:
`ax^2 + bx + c = 0`

From your equation:
a= 1
b=5
c=7

∴ 25 - 4 x 1 x 7
= -3

A negative discriminant means there is no real root.
If you wanted to go further, it would have complex x roots. But yeah:

`x^2+5x+7=0` has no real roots therefore no defined solutions.