If a- b-c-d- then the equation x2-ax-3bx2-cx-bx2-dx-2b-0 has

The discriminants of the given quadratic factors are D_1=a^2+12b, D_2=c^2 4b and D_3=d^2 8b nbsp; D_1+D_2+D_3=a^2+c^2+d^2≥0 Hence, at least one of D_1,D_2,D_3 ...

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Standard XII

Mathematics

Discriminant

If a, b,c,d∈ℝ...

Question

If $a, b, c, d \in R,$ then the equation $(x^{2} + a x - 3 b) (x^{2} - c x + b) (x^{2} - d x + 2 b) = 0$ has

no real roots

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at least two real roots

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at least four real roots

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exactly six real roots

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Solution

The correct option is B at least two real roots
The discriminants of the given quadratic factors are
$D_{1} = a^{2} + 12 b, D_{2} = c^{2} - 4 b$ and $D_{3} = d^{2} - 8 b$
$D_{1} + D_{2} + D_{3} = a^{2} + c^{2} + d^{2} \geq 0$
Hence, at least one of $D_{1}, D_{2}, D_{3}$ is non-negative.
Therefore, the equation has at least two real roots.

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If a,b,c,d∈R, then the equation (x2+ax−3b)(x2−cx+b)(x2−dx+2b)=0 has

The correct option is B at least two real roots The discriminants of the given quadratic factors are D1=a2+12b,D2=c2−4b and D3=d2−8b D1+D2+D3=a2+c2+d2≥0 Hence, at least one of D1,D2,D3 is non-negative. Therefore, the equation has at least two real roots.

If $a, b, c, d \in R,$ then the equation $(x^{2} + a x - 3 b) (x^{2} - c x + b) (x^{2} - d x + 2 b) = 0$ has