Question

How do you find the number of solutions using the discriminant?

Answer 1

Hint: First we will reduce the equation further if possible. Then we will try to factorise the terms in the equation. Then we will use the quadratic formula to solve for the value of $ x $ using the formula which is given by $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ .

Complete step-by-step answer:
We generally start off by taking all the terms to one side. Now, for the quadratic formula which is also called as the discriminant for $ a{x^2} + bx + c = 0 $ which is given by
 $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ .
The discriminant is the portion of the quadratic equation within the radical which is given by:
 $ {b^2} - 4ac $ .
Now the type of solution and the number of solutions is decided by the sign of the discriminant portion of the quadratic equation within the radical.
Now if the sign of the discriminant is positive, we will be getting two real solutions. If the value of the discriminant comes out as zero then you will get only one solution. And then finally if you get negative values then you will get complex solutions.

Note: While comparing the values of the given equation with the general form of quadratic equation which is given by $ a{x^2} + bx + c = 0 $ , compare along with their respective signs. While applying the quadratic formula, make sure you substitute all the values along with their respective signs. Solve all the equations separately, so that you don’t miss any term of the solution. Check if the solution satisfies the original equation completely. If any term of the solution doesn’t satisfy the equation, then that term will not be considered as a part of the solution.