Quadratic formula

Consider the quadratic equation ax^{2} + 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

Therefore,

So, the roots of ax^{2} + bx + c = 0 are , if b^{2} – 4ac ≥ 0. If b^{2} –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:

Solution:

Therefore,

So, the roots are 2/3 and 1.

(ii) x^{2} + 4x + 5 = 0. Here, a = 1, b = 4, c = 5. So, b^{2} – 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,

Therefore,

So, the roots are