Question

Which of the following equations has no real roots?

Options

• `x^2 - 4x + 3sqrt(2) = 0`

• `x^2 + 4x - 3sqrt(2) = 0`

• `x^2 - 4x - 3sqrt(2) = 0`

• `3x^2 + 4sqrt(3)x + 4 = 0`

Answer 1

`bb(x^2 - 4x + 3sqrt(2) = 0)`

Explanation:

(A) The given equation is `x^2 - 4x + 3sqrt(2)` = 0

On comparing with ax² + bx + c = 0, we get

a = 1, b = – 4 and c = `3sqrt(2)`

The discriminant of `x^2 - 4x + 3sqrt(2)` = 0 is 

D = b² – 4ac

= `(-4)^2 - 4(1)(3sqrt(2))`

= `16 - 12sqrt(2)`

= 16 – 12 × (1.41)

= 16 – 16.92

= – 0.92

⇒ b² – 4ac < 0

(B) The given equation is `x^2 + 4x - 3sqrt(2)` = 0

On comparing the equation with ax² + bx + c = 0, we get

a = 1, b = 4 and c = `-3sqrt(2)`

Then, D = b² – 4ac

= `(-4)^2 - 4(1)(-3sqrt(2))`

= `16 + 12sqrt(2) > 0`

Hence, the equation has real roots.

(C) Given equation is `x^2 - 4x - 3sqrt(2)` = 0

On comparing the equation with ax² + bx + c = 0, we get

a = 1, b = – 4 and c = `-3sqrt(2)`

Then, D = b² – 4ac

= `(-4)^2 - 4(1)(-3sqrt(2))`

= `16 + 12sqrt(2) > 0`

Hence, the equation has real roots.

(D) Given equation is `3x^2 + 4sqrt(3)x + 4` = 0

On comparing the equation with ax² + bx + c = 0, we get

a = 3, b = `4sqrt(3)` and c = 4

Then, D = b² – 4ac

= `(4sqrt(3))^2 - 4(3)(4)`

= 48 – 48

= 0

Hence, the equation has real roots.

Hence, `x^2 - 4x + 3sqrt(2)` = 0 has no real roots.