If the equation a(b-c) x²+b(c-a) x+c(a-b)=0 has equal roots, where a+c=15 and b=36/5, then a²+c² is equal to _________

Solution

Maths sir · 4 steps

The coefficients add up to zero, so 1 is a root; equal roots means both roots are 1.

1. Spot a root. Add the three coefficients: ab - ac + bc - ab + ca - bc = 0. When the coefficients sum to zero, putting x = 1 gives zero, so x = 1 is a root.
2. Use equal roots. The roots are equal and one of them is 1, so both roots are 1. Their product is then 1, and the product of roots is c(a-b)/a(b-c).

   c(a-b) = a(b-c) ⇒ 2ac = b(a+c)

3. Find ac. 2ac = 36/5 × 15 = 108, so ac = 54. In other words, b is the harmonic mean of a and c.
4. So. a² + c² = (a+c)² - 2ac = 225 - 108 = 117.

The common slip: 9. Writing ac = b(a+c) = 108 and dropping the factor 2 gives a² + c² = 225 - 216 = 9, not 117.

The idea to keep. Whenever the coefficients of a quadratic add up to zero, x = 1 is a root. With equal roots, the other root is also 1, so the product of roots must be 1.

Answer from the NTA final answer key. Solution by PrepWiser.