a, b, c, d are real numbers.

Question

a, b, c, d are real numbers. If $D ₁ a n d D ₂$ are discriminants of the quadratic equations $x ² + a x + b = 0 a n d x ² + c x + d = 0$ respectively, and ac = 2(b + d), then which of the following statements is true:

(i) At least one $D ₁ o r D ₂$ is greater than or equal to zero.
(ii) Both $D ₁ a n d D ₂$ are greater than zero.
(iii) At least one of the two equations has real roots.
(iv) Both the equations have real roots.

Choose the correct option from the following:

Smart Achievers · Accepted Answer

(1)

$D ₁ = a ² - 4 b a n d D ₂ = c ² - 4 d$

$N o w, D ₁ + D ₂ = a ² + c ² - 4 (b + d) = a ² + c ² - 2 a c [S i n c e 2 (b + d) = a c]$

$\Rightarrow D ₁ + D ₂ = (a - c) ² \geq 0$

So, at least one of $D ₁ a n d D ₂$ is greater than or equal to zero. Thus, at least one of the given two equations has real roots.

Solution

1 (i) and (iii)

2 (i) and (iv)

3 (ii) and (iii)

4 (ii) and (iv)

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