Question

determine k so that the eq. x2 -4x+k=0 has ( i ) two distinct roots (ii) coincident roots​

Answer 1

✪ANSWER✪

1. k ≠ 4 or k ∈ (-∞, 4)∪(4 , ∞)
2. k = 4

★EXPLAINATION★

☆GIVEN☆

• Quadretic Equation $x² - 4x + k = 0$

☆TO FIND☆

• Value of k for two distinct roots.
• Value of k for co-incident roots.

☆CALCULATION☆

Discriminant of the quadratic equation is given by

$D = b² - 4ac$

$D = (-4)² - 4.1.k$

$D = 16 - 4k$

Distinct Roots

• Real and distinct roots

Roots of a quadretic equation(with real co-efficients) are real and distinct when D > 0.

$⇒ 16 - 4k > 0$

$⇒ 16 > 4k$

$⇒ \frac{16}{4} > k$

$⇒ k < 4$

• Complex and distinct roots

Roots of a quadretic equation(with real co-efficients) are complex and distinct when D < 0.

$⇒ 16 - 4k < 0$

$⇒ 16 < 4k$

$⇒ \frac{16}{4} < k$

$⇒ k > 4$

In both cases, roots are distinct. Hence value of k for distinct roots is

k > 4 or k < 4 i.e. k ≠ 4

Coincident Roots

Roots of a quadretic equation(with real co-efficients) are real and equal when D = 0.

$⇒ 16 - 4k = 0$

$⇒ 16 = 4k$

$⇒ \frac{16}{4} = k$

$⇒ k = 4$

Hence, the roots are coincident when k = 4.