Question

Write the condition to be satisfied for which equations ax² + 2bx + c = 0 and bx²-2√acx+b=0 have equal roots.

Answer 1

SOLUTION :  

Given : ax² + 2bx + c = 0 …………(1)

and bx² - 2√acx + b = 0…………..(2)

On comparing the given equation with Ax² + Bx + C = 0  

Let D1 & D2 be the discriminants of the two given equations .

For eq 1 :  

Here, A = a  , B =  2b , C = c

D(discriminant) = B² – 4AC

D1 = (2b)² - 4 × a × C

D1 = 4b² - 4ac ………(3)

For eq 2 :  

bx² - 2√acx + b = 0

Here, A = b  , B =  - 2√ac, C = b

D(discriminant) = B² – 4AC

D2 = (- 2√ac)² - 4 × b × b  

D2 = 4ac - 4b² …………(4)

Given roots are  equal for both the equations  so, D1 & D2 = B² – 4AC = 0

D1 = 0  

4b² - 4ac = 0  

[From eq 3]

4b²  = 4ac  

b² = ac  ………….(5)

D2 = 0

4ac - 4b² =  0

4ac  = 4b²

ac  = b² …………(6)

From eq 5 & 6 ,  

b² = ac  

Hence, b² = ac is the condition under which the given equations have equal roots.

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Answer 2

Solution :

Given two quadratic equations

ax² + 2bx + c = 0 ,

and

bx² - 2√acx + b = 0

Discreminant (D) = 0

[ Since , Equation has equal

roots ]

(2b)² - 4ac = (-2√ac)² - 4b²

=> 4b² - 4ac = 4ac - 4b²

=> Divide each term by 4 ,

we get

b² - ac = ac - b²

=> 2b² = 2ac

=> b² = ac

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