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प्रश्न
Write the condition to be satisfied for which equations ax2 + 2bx + c = 0 and \[b x^2 - 2\sqrt{ac}x + b = 0\] have equal roots.
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उत्तर
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,
`D1 = (2b)^2 - 4ac`
= `4b^2 - 4ac`
And `D_= (-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.
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