If x 2- b 1 x - c 1-0 and x 2- b 2 x - c 2-0 are the two quadratic equations such that b 1 b 2-2 c 1- c 2 and b 1- b 2- c 1- c 2- R- then whichof the following is always true-A- at least one equation has imaginary rootsB- at least one equation has real rootsC- exactly one equation has real rootsD- both the equations have real roots

The correct option is B at least one equation has real roots nbsp;Let D_1 and D_2 be discriminants of nbsp; x^2+b_1x+c_1=0 and x^2+b_2x+c_2=0, respectively ...

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Byju's Answer

Standard XIII

Mathematics

Discriminant

If x 2+ b 1 x...

Question

If $x^{2} + b_{1} x + c_{1} = 0$ and $x^{2} + b_{2} x + c_{2} = 0$ are the two quadratic equations such that $b_{1} b_{2} = 2 (c_{1} + c_{2})$ and $b_{1}, b_{2}, c_{1}, c_{2} \in R,$ then which of the following is always true ?

at least one equation has imaginary roots

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at least one equation has real roots

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exactly one equation has real roots

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both the equations have real roots

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Solution

The correct option is B at least one equation has real roots
Let $D_{1}$ and $D_{2}$ be discriminants of $x^{2} + b_{1} x + c_{1} = 0$ and $x^{2} + b_{2} x + c_{2} = 0,$ respectively. Then, $D_{1} + D_{2} = b_{1}^{2} - 4 c_{1} + b_{2}^{2} - 4 c_{2}$
$= (b_{1}^{2} + b_{2}^{2}) - 4 (c_{1} + c_{2})$
$= b_{1}^{2} + b_{2}^{2} - 2 b_{1} b_{2}$
$= (b_{1} - b_{2})^{2} \geq 0$
$\Rightarrow$ At least one of $D_{1}, D_{2}$ is non-negative.
Hence, at least one of the equations has real roots.

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If x2+b1x+c1=0 and x2+b2x+c2=0 are the two quadratic equations such that b1b2=2(c1+c2) and b1,b2,c1,c2∈R, then which of the following is always true ?

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