Solve the following questions.
1) If 2 is a root of the quadratic equation 3x" + px - 8 = 0 and the quadratic equation 4x” – 2px + k = 0 has real and equal roots, find k. 2
Answers
EXPLANATION.
2 is a root of the quadratic equation 3x² + px - 8 = 0.
Quadratic equation 4x² - 2px + k = 0 has real and equal roots.
As we know that,
Put the value of x = 2 in equation, we get.
⇒ 3x² + px - 8 = 0.
⇒ 3(2)² + p(2) - 8 = 0.
⇒ 3(4) + 2p - 8 = 0.
⇒ 12 + 2p - 8 = 0.
⇒ 4 + 2p = 0.
⇒ 2p = - 4.
⇒ p = - 2.
Put the values of the equation (2), we get.
⇒ 4x² - 2px + k = 0.
⇒ 4x² - 2(-2)(x) + k = 0.
⇒ 4x² + 4x + k = 0.
For real and equal roots.
⇒ D = 0 Or b² - 4ac = 0.
⇒ (4)² - 4(4)(k) = 0.
⇒ 16 - 16k = 0.
⇒ 16k = 16.
⇒ k = 1.
Values of k = 1.
MORE INFORMATION.
Nature of the roots of quadratic expression.
(1) Roots are real and unequal, if b² - 4ac > 0.
(2) Roots are rational and different, if b² - 4ac is a perfect square.
(3) Roots are real and equal, if b² - 4ac = 0.
(4) If D < 0 Roots are imaginary and unequal Or complex conjugate.