Question

A ,b, c, d be distinct real numbers and a and b are the roots of the quadratic equation x^2-2cx-5d=0. If c and d are the roots of quadratic equation x^2-2ax-5b=0 then find the numerical value of a + b + c + d

Answer 1

Given info : a ,b, c, d be distinct real numbers and a and b are the roots of the quadratic equation x² - 2cx -5d=0. If c and d are the roots of quadratic equation x² - 2ax- 5b=0

To find : the numerical value of a + b + c + d = ?

solution : a and b are roots of x² - 2cx - 5d = 0

so, sum of roots = a + b = -(-2c)/1 = 2c ...(1)

similarly, c and d are roots of x² - 2ax - 5b = 0

so sum of roots = c + d = -(-2a)/1 = 2a ...(2)

adding equations (1) and (2) we get,

a + b + c + d = 2(c + a) ...(3)

now we have to find c + a , don't we ?

subtracting equation (2) from equation (1) we get,

a + b - c - d = 2(c - a)

⇒b - d = 3(c - a)

⇒d - b = 3(a - c) ....(4)

putting a in x² - 2cx - 5d = 0 .

a² - 2ca - 5d = 0 ....(5)

putting c in x² - 2ax - 5b = 0

c² - 2ac - 5b = 0 ...(6)

subtracting eq (6) from eq (5) we get,

(a² - c²) - 5(d - b) = 0

from equation (4) , (a - c)(a + c) - 5 × 3(a - c) = 0

(a - c)[(a + c) - 15] = 0

a + c = 15 now put this in eq (3) we get,

a + b + c + d = 2 × 15 = 30