Question

Write the condition to be satisfied for which equations ax² + 2bx + c = 0 and \[bx^2 - 2\sqrt{ac}x + b = 0\] have equal roots.

Answer 1

The given equations are

 ax² + 2bx + c = 0  …... (1)

And \[b x^2 - 2\sqrt{ac}x + b = 0\] …… (2)

roots are equal.

Let D₁ and D₂ be the discriminants of equation (1) and (2) respectively,

Then,

`D_1 = (2b)^2 - 4ac`

= `4b^2 - 4ac`

And `D_2 = (-2sqrtac)^2 - 4 xx b xx b `

  ` = 4ac - 4b^2`

Both the given equation will have real roots, if `D_1 ≥ 0" and "D_2 ≥ 0`

`4b^2 - 4ac ≥ 0`

`4b^2 ≥ 4ac`

`b^2 ≥ ac`…… (3)

`4ac - 4b^2 ≥ 0`

`4ac ≥ 4b^2`

`ac ≥ b^2` …... (4)

From equations (3) and (4) we get

b² = ac

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