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Quadratic formula

Consider the quadratic equation ax2 + bx + c = 0 (a ≠ 0). Dividing throughout by a, we get
quadratic formula to find roots of the equation This is the same as
quadratic formula So, the roots of the given equation are the same as those of
quadratic formula to find roots of the equation If quadratic formula , then by taking the square roots in (1), we get
quadratic formula to find roots of the equation Therefore,
quadratic formula So, the roots of ax2 + bx + c = 0 are roots of quadratic equations , if b2 – 4ac ≥ 0. If b2 –4ac < 0, the equation will have no real roots.

Thus, if quadratic formula , then the roots of the quadratic equation ax2 + bx + c = 0 are given by roots of quadratic equations by quadratic formula .

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,
roots of equation of quadratic formula which is a quadratic equation in x.
Using the quadratic formula, we get
quadratic formula 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:
roots of quadratic equation by quadratic formula
Solution:
quadratic formula Therefore, 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 quadratic formula will not have any real value.
So, there are no real roots for the given equation.
quadratic formula So,
quadratic formula Therefore,
quadratic formula So, the roots are quadratic formula

Next Topic : Nature of roots
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