Important Concepts and Formulas
1. An equation of the form ax2
+ bx + c = 0, where a, b, c are real numbers and a ≠ 0 is called a quadratic equation.
2. A real number α is said to be a root of the quadratic
equation ax2
+ bx + c = 0, if aα2 + bα + c = 0. The zeroes of the quadratic
polynomial ax2 + bx + c and the roots of the quadratic
equation ax2 + bx + c = 0 are the same.
3. If we can factorize ax2
+ bx + c, a ≠ 0, into product of two linear
factors, then the roots of the quadratic equation ax2 + bx + c
= 0 can be found by equating each factor to zero. For example, if (x + α) and (x + β) are the factors of quadratic
equation ax2 + bx + c = 0, then the roots are x = -
α and x = - β.
4. A quadratic equation can also be solved by the
method of completing the square. For example, we can make the quadratic
equation in the form a2 + 2ab + b2 and write it as (a +
b)2.
5. We can also find the roots of the quadratic
equation using the quadratic formula which is given by:
, provided that b2 – 4ac ≥ 0.
This is also called Sreedharacharya formula.
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6. A quadratic equation ax2
+ bx + c, a ≠ 0 has
I.
two
distinct real roots, if b2 – 4ac ˃ 0,
II.
two
equal roots, if b2 – 4ac = 0, and
III.
no
real roots, if b2 – 4ac < 0.