#### Question

If a and b can take values 1, 2, 3, 4. Then the number of the equations of the form ax^{2} +bx + 1 = 0 having real roots is

10

7

6

12

#### Solution

Given that the equation `ax^2 +bx +1 = 0`_{.}

For given equation to have real roots, discriminant (D) ≥ 0

⇒ b^{2} − 4a ≥ 0

⇒ b^{2} ≥ 4a

⇒ b ≥ 2√a

Now, it is given that a and b can take the values of 1, 2, 3 and 4.

The above condition b ≥ 2√a can be satisfied when

i) b = 4 and a = 1, 2, 3, 4

ii) b = 3 and a = 1, 2

iii) b = 2 and a = 1

So, there will be a maximum of 7 equations for the values of (a, b) = (1, 4), (2, 4), (3, 4), (4, 4), (1, 3), (2, 3) and (1, 2).

Is there an error in this question or solution?

#### APPEARS IN

Solution If a and B Can Take Values 1, 2, 3, 4. Then the Number of the Equations of the Form Ax2 + Bx + 1 = 0 Having Real Roots is Concept: Solutions of Quadratic Equations by Factorization.