Question

find roots in equation.

Answer 1

given eqn is, 4x^2 - 4px + (p^2-q^2) = 0
compare the given eqn with ax^2 + bx + c = 0

=> a = 4, b = -4p, c= (p^2-q^2)
now let x1, x2 be the roots.
wkt, sum of roots = -b/a and product of the roots = c/a

=> sum of roots = x1 + x2 = -b/a = -(-4p)/4 = p
=> x1+x2 = p ----eqn (1)
product of the roots = x1.x2 = c/a = (p^2-q^2)/4

(x1-x2)^2 = (x1+x2)^2 - 4x1.x2
= p^2 - 4 (p^2-q^2)/4
= p^2 - (p^2-q^2)
= p^2 - p^2+q^2)
(x1-x2)^2 = q^2
=> x1-x2 = q---eqn (2)

add eqn (1) & eqn (2)
=> (x1+x2)+(x1-x2) = p+q
=> 2x1 = p+q
=> x1 = (p+q)/2

substituting x1 value in eqn (1) and solving it
=> (p+q)/2 + x2 = p
=> x2 = (p-q)/2

so, the roots are (p+q)/2 and (p-q)/2.

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