Write the condition to be satisfied for which equations ax² + 2bx + c = 0 and have equal roots.
Answers
Write the condition to be satisfied for which equations ax² + 2bx + c = 0 and have equal roots.
The complete question is:
Write the condition to be satisfied for which equations ax² + 2bx + c = 0 and bx^2 - 2√ac x + b = 0 have equal roots.
b^2 = ac is the condition to be satisfied for which equations ax² + 2bx + c = 0 and bx^2 - 2√ac x + b = 0 have equal roots.
Given equations:
ax^2 + 2bx + c = 0
bx^2 - 2√ac x + b =0
Discriminant is given by, D = b^2 - 4ac.
Let D1 and D2 be the discriminant of given equations respectively.
D1 = (2b)^2 - 4ac = 4b^2 - 4ac
D2 = (-2√ac)^2 - 4bb = 4ac - 4b^2
Both of the equations will have equal roots, if
D1 ≥ 0 and D2 ≥ 0
4b^2 - 4ac ≥ 0 and 4ac - 4b^2 ≥ 0
4b^2 ≥ 4ac and 4ac ≥ 4b^2
b^2 ≥ ac and ac ≥ b^2
From the above conditions, we get,
b^2 = ac.
Therefore, b^2 = ac is the condition to be satisfied for which equations ax² + 2bx + c = 0 and bx^2 - 2√ac x + b =0 have equal roots.