Nature of Roots of Quadratic Equations [Examples]
〉〉 Without solving, comment upon the nature of the roots for following 3 Equations.
Example 1: 6x² – 13x + 4 = 0
From ax² + bx + c=0; by comparing, we get a = 6, b = -13, c = 4;
So coefficients are real.
Discriminant ‘D’= b² – 4 a c ⇒ (-13)² -4 (6)(4) ⇒ 169 -96 ⇒ 76>0
Hence the roots are Real and Unequal.
Example 2: 4x² – 12x + 9 = 0
From ax² + bx + c=0; by comparing, we get a = 4, b = -12, c = 9;
So coefficients are real.
Discriminant ‘D’= b² – 4 a c ⇒ (-12)² -4 (4)(9) ⇒ 144 -144 = 0
Hence the roots are Real and Equal.
Example 3: 2x² + 8x + 9 = 0
From ax² + bx + c=0; by comparing, we get a = 2, b = 8, c = 9;
So coefficients are real.
Discriminant ‘D’= b² – 4 a c ⇒ (8)² -4 (2)(9) ⇒ 64 -72 ⇒ -8<0
Hence the roots are Imaginary.
Example 4: Find the value of K such that the quadratic equation kx² – 7x + 2 = 0 has equal roots!
From given equation kx² – 7x + 2=0; on comparing with ax² + bx + c=0; we get a = k, b = -7, c = 2;
Since roots are equal. so Discriminant ‘D’ = 0
b² – 4 a c = 0
(-7)² -4 (k)(2)
49 – 8k = 0
8k = 49
Answer: k = 49/8
Hope concept of nature of roots is better clear to you.
You can attempt quiz on Equations here BasicLevel