We have learnt different types of polynomials. One type was the quadratic polynomial of the form ax2 + bx + c, a ≠ 0. When we equate this polynomial to zero, we get a quadratic equation. Quadratic equations come up when we deal with many real-life situations.
Here we will study quadratic equations, and various ways of finding their roots.
A quadratic equation in the variable x is an equation of the form ax2 + bx + c = 0, where a, b, c are real numbers, a ≠ 0.
For example, 2x2 + x — 300 = 0 is a quadratic equation.
Consider the quadratic equation 2x2 —3x + 1 = 0. If we replace x by 1 on the LHS of this equation, we get (2 x 12) — (3 x 1) + 1 = 0 = RHS of the equation.
When a real number a is called a root of the quadratic equation ax2 + bx + c = 0, a ≠ 0 if a a2 + ba + c = 0. We also say that x = a is a solution of the quadratic equation, or that a satisfies the quadratic equation. Note that the zeroes of the quadratic polynomial ax2 + bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same.
We have learnt one method of obtaining the roots of a quadratic equation. In this section, we learn another method.
if b2 —4ac ≥ 0, then the roots of the quadratic equation ax2 + bx + c = 0 are given by
- b ±√b2 - 4ac/2a
This formula for finding the roots of a quadratic equation is known as the quadratic formula.
If b2 −4ac < 0, then there is no real number whose square is b2 −4ac. Therefore, there are no real roots for the given quadratic equation in this case.
Since b2 −4ac determines whether the quadratic equation ax2 + bx + c = 0 has real roots or not, b2 − 4ac is called the discriminant of this quadratic equation.
So, a quadratic equation ax2 + bx + c = 0 has
Two distinct real roots, if b2 −4ac > 0,
Two equal real roots, if b2 −4ac = 0,
No real roots, if b2 − 4ac < 0.