Question

If the equation ax^2 + x + b = 0 and x^2 + bx + a = 0 a have common root and the second Equation has equal roots, then 2a^2 + b​

Answer 1

Answer:

Let u be the common root of the two quadratics.

Since x² + bx + a = 0 has equal roots, that is, both of its roots are u, we have

• sum of roots = 2u = -b
• product of roots = u² = a

Since u is a root of ax² + x + b = 0, we also have

• au² + u + b = 0.

Multiplying this last equation by 2 gives

• 2au² + 2u + 2b = 0.

Now substituting the values above for 2u and u² gives

• 2a×a + (-b) + 2b = 0    ⇒    2a² + b = 0.

Hope that helps!