Quadratic Equations Introduction
Definition
- A quadratic is a polynomial whose highest degree is 2.
- aX2 + bX + c = 0 where a, b, c are numbers and a ≠ 0. [Standard Form of Quadratic equation]
- The co-efficient of x2 is called the leading co-efficient.
- Root : A solution of a quadratic equation is known as Root. There is always two roots of a quadratic equations it may be equal, different or complex.
Quadratic Equation
Let aX2 + bX + c = 0 where a, b and c are real numbers and a ≠ 0.
Number of roots
Discriminant : In Qudratic formula (b2 - 4ac) is called the Discriminant, where a, b and c are real numbers, a ≠ 0. It is represented by D.
- When D = b2 - 4ac is positive, then the roots are real and unequal.
- When D = b2 - 4ac is zero, then roots are real and equal.
- When D = b2 - 4ac is negative, then roots are unequal and imaginary.
- When D = b2 - 4ac is positive and perfect square, then the roots are real,rational and unequal.
- When D = b2 - 4ac is positive but not perfect square, then roots are real, irrational and unequal.
- When D = b2 - 4ac is perfect square and a or b is irrational, then roots are irrational.
1. Find the number of roots of x2 + 3x - 10 = 0
Discriminant, D = b2 - 4ac
$=9+40=49 >0 $
Here, D > 0 and perfect square. Hence the equation will have real, rational and unequal roots.
Let's find the root by solving the quadratic equation.
The solution is $\text{x}=2,-5$
2. Find the number of roots of x2 - 6x + 9 = 0
Discriminant, D = b2 - 4ac
$ =36-36=0 $
Here, D = 0. Hence the equation will have one real and equal roots.
Let's find the root by solving the quadratic equation.
The solution is $\text{x}=3$
3. Find the number of roots of x2 + 3 =0
Discriminant, D = b2 - 4ac
Here, D < 0. Hence the equation will have unequal and imaginary.
Let's find the root by solving the quadratic equation.
Sum and Products of the roots
Let p and q are the roots of a quadratic equation ax2 + bx + c = 0 where a ≠ 0
Sum of the roots = p + q = − b/a
Product of the roots = pq = c/a
Example : Find the sum and product of the roots for the given equation x2 + 3x - 10 = 0
Let p and q are the roots of the given equation, then Sum of the roots = p + q = − b/a = -3/1 = -3 and Product of the roots = pq = c/a = -10/1 = -10