Write the condition to be satisfied for which equations
Question:

Write the condition to be satisfied for which equations $a x^{2}+2 b x+c=0$ and $b x^{2}-2 \sqrt{a c} x+b=0$ have equal roots.

Solution:

The given equations are

$a x^{2}+2 b x+c=0 \ldots \ldots$ (1)

And, $b x^{2}-2 \sqrt{a c} x+b=0$……(2)

roots are equal.

Let $D_{1}$ and $D_{2}$ be the discriminants of equation (1) and (2) respectively,

Then,

$D_{1}=(2 b)^{2}-4 a c$

$=4 b^{2}-4 a c$

And $D_{2}=(-2 \sqrt{a c})^{2}-4 \times b \times b$

$=4 a c-4 b^{2}$

Both the given equation will have real roots, if $D_{1} \geq 0$ and $D_{2} \geq 0$

$4 b^{2}-4 a c \geq 0$

$4 b^{2} \geq 4 a c$

$b^{2} \geq a c \ldots \ldots$ (3)

$4 a c-4 b^{2} \geq 0$

$4 a c \geq 4 b^{2}$

$a c \geq b^{2} \ldots \ldots(4)$

From equations (3) and (4) we get

$b^{2}=a c$

Hence, $b^{2}=a c$ is the condition under which the given equations have equal roots.

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