Question 2:

Find the values of k for each of the following quadratic equations, so that they have two equal roots.

(I) 2x² + kx + 3 = 0    (II) kx (x − 2) + 6 = 0

Answer:

We know that if an equation ax² + bx + c = 0 has two equal roots, its discriminant

(b² − 4ac) will be 0.

(I) 2x² + kx + 3 = 0

Comparing equation with ax² + bx + c = 0, we obtain a = 2, b = k, c = 3

Discriminant = b² − 4ac = (k)²− 4(2) (3) = k² − 24

For equal roots,

Discriminant = 0

k² − 24 = 0

k² = 24

k = ±√24 = ±2√6

(II) kx (x − 2) + 6 = 0

or kx² − 2kx + 6 = 0

Comparing this equation with ax² + bx + c = 0, we obtain a = k, b = −2k, c = 6

Discriminant = b² − 4ac = (− 2k)² − 4 (k) (6) = = 4k² − 24k

For equal roots,

b² − 4ac = 0

4k² − 24k = 0

4k (k − 6) = 0

Either 4k = 0 or k = 6 = 0

k = 0 or k = 6

However, if k = 0, then the equation will not have the terms ‘x²’ and ‘x’.

Therefore, if this equation has two equal roots, k should be 6 only.

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