Consider the quadratic equation ax2 + bx + c = 0 (a ≠ 0). Dividing throughout by a, we get
This is the same as
So, the roots of the given equation are the same as those of
If , then by taking the square roots in (1), we get
So, the roots of ax2 + bx + c = 0 are , if b2 – 4ac ≥ 0. If b2 –4ac < 0, the equation will have no real roots.
Thus, if , then the roots of the quadratic equation ax2 + bx + c = 0 are given by .
This formula for finding the roots of a quadratic equation is known as the quadratic formula.
Example: Find two consecutive odd positive integers, sum of whose squares is 290.
Solution: Let the smaller of the two consecutive odd positive integers be x. Then, the second integer will be x + 2. According to the question,
which is a quadratic equation in x.
Using the quadratic formula, we get
But x is given to be an odd positive integer. Therefore, x ≠ – 13, x = 11.
Thus, the two consecutive odd integers are 11 and 13.
Check : 112 + 132 = 121 + 169 = 290.
Example: Find the roots of the following quadratic equations, if they exist, using the quadratic formula:
So, the roots are 2/3 and 1.
(ii) x2 + 4x + 5 = 0. Here, a = 1, b = 4, c = 5. So, b2 – 4ac = 16 – 20 = – 4 < 0.
Since the square of a real number cannot be negative, therefore will not have any real value.
So, there are no real roots for the given equation.
So, the roots are