How can you use the discriminant to find out the nature of the roots of `7x^2=19x` ?

Given a quadratic equation `ax^2+bx+c = 0`, the discriminant is the expression `b^2-4ac`. Evaluating the discriminant and observing its sign can show how many real solutions the equation has.

• If `b^2-4ac > 0`, then there are `2` solutions.
• If `b^2-4ac = 0`, then there is `1` solution.
• If `b^2-4ac < 0`, then there are `0` solutions.

The reasoning behind this is clear upon looking at the quadratic formula:

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

If the discriminant is positive, then two distinct answers will be obtained by adding or subtracting its square root. If it is `0`, then adding or subtracting will lead to the same single answer. If it is negative, it has no real square root.

Putting the given equation in standard form, we have

`7x^2 = 19x`

`=> 7x^2 - 19x + 0 = 0`

`=> b^2-4ac = (-19)^2 -4(7)(0) = 19^2 > 0`

Thus, there are two solutions.

We can also find whether the solution(s) will be rational or irrational. If the discriminant is a perfect square, then its square root will be an integer, making the solutions rational. Otherwise, the solutions will be irrational.

In the above case, `b^2-4ac = 19^2` is a perfect square, meaning the solutions will be rational.

(If we solve the given equations, we will find the solutions to be `x=0` or `x=19/7`)