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

1 Answer
Oct 26, 2016

Answer:

Check the sign and value of the discriminant to find that the given equation will have two rational solutions.

Explanation:

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#)