If one root of the quadratic equation kx2 – 20x + 34 = 0 is 5 – 2√2 , find k.

7. If one root of the quadratic equation kx² – 20x + 34 = 0 is 5 – 2√2 , find k.

Sol. kx² – 20x + 34 = 0

Comparing with ax² + bx + c = 0 we have a = k, b = – 20, c = 34

Let  α  and β be the roots of given quadratic equation.

∴ α  = 5 – 2√2 [Given]

∴  β  = 5 + 2 √2

∴  α  + β  = – b/a = –(– 20)/k = 20/k

∴  5 – 2√2 + 5 + 2√2 = 20/k

∴  10 = 20/k

∴ k = 20/10

∴  k = 2

If one root of the quadratic equation kx2 – 5x + 2 = 0 is 4 times the other, find k.

1. If one root of the quadratic equation kx² – 5x + 2 = 0 is 4 times the

other, find k.

Sol. kx² – 5x + 2 = 0

Comparing with ax² + bx + c = 0 we have a = k, b = – 5, c = 2

Let α  and β  be the roots of given quadratic equation.

α = 4 β   [Given]

∴ α +β = ^{- b}/_a = ^{- (- 5)}/_k = ⁵/_k

∴ 4β + β = 5/k

∴ 5β = 5/k

∴ β = 5/5k

∴ β = 1/k

Also,  α.β = c/a = 2/k

∴ 4β .β = 2/k

∴ 4(1/k)(1/k) = 2/k

∴ 4/k² =  2/k

∴ 4/2 = k²/k

∴ 2 = k