Find the values of k for which the following equation has equal roots :
(k−12)x 2 +2(k−12)x+2=0
Answers
EXPLANATION.
Quadratic equations,
⇒ (k - 12)x² + 2(k - 12)x + 2 = 0.
As we know that,
D = Discriminant Or b² - 4ac.
Roots are real and equal,
D = 0 Or b² - 4ac = 0.
⇒ [2(k - 12)²] - 4(k - 12)(2) = 0.
⇒ [4(k² + 144 - 24k)] - 4[2k - 24] = 0.
⇒ [4k² + 576 - 96k] - 8k + 96 = 0.
⇒ 4k² + 576 - 96k - 8k + 96 = 0.
⇒ 4k² - 104k + 672 = 0.
⇒ 4[k² - 26k + 168] = 0.
⇒ k² - 26k + 168 = 0.
⇒ k² - 14k - 12k + 168 = 0.
⇒ k(k - 14) - 12(k - 14) = 0.
⇒ (k - 12)(k - 14) = 0.
⇒ k = 12 and k = 14.
MORE INFORMATION.
Nature of the factors of the quadratic expression.
(1) = Real and different, if b² - 4ac > 0.
(2) = Rational and different, if b² - 4ac is a perfect square.
(3) = Real and equal, if b² - 4ac = 0.
(4) = If D < 0 Roots are imaginary and unequal or complex conjugate.
a = k – 12, b = 2 (k – 12), c = 2
∴ D = b2 – 4ac = [2(k– 12)]2
– 4(k – 12) x 2
= 4 (k – 12)2 – 8 (k– 12)
Roots are equal, if D = 0
⇒ 4 (k – 12) 2 – 8 (k – 12) = 0
⇒ 4(k – 12)(k – 12 – 2) = 0
⇒ (k – 12) (k – 14) = 0
⇒ k = 12 or 14
But k = 12 does not satisfy the eqn.
k = 14 Ans.
