Question

What will be the value of a, for which the equation 5x^2 + ax + 5 and x^2 – 12x + a will have real roots?

(a) a = 37

(b) 10 < a < 36

(c) 36 < a < 10

(d) a = 9

The question was posed to me during an online interview.

My doubt stems from Quadratic Equation in division Quadratic Equation of Mathematics – Class 10

Answer 1

Right option is (b) 10 < a < 36

The explanation is: The roots of both the equations are real.

Discriminant of 5x^2 + ax + 5 : b^2 – 4ac = a^2 – 4 × 5 × 5 = a^2 – 100

Since, roots are real; discriminant will be greater than 0.

a^2 ≥ 100

a ≥ ±10

Now, discriminant of x^2 – 12x + a : b^2 – 4ac = -12^2 – 4 × 1 × a = 144 – 4a

Since, roots are real; discriminant will be greater than 0.

144 ≥ 4a

a ≤ \(\frac {144}{4}\) = 36

For both the equations to have real roots the value of a must lie between 36 and 10.