For the quadratic equation ax 2- bx -4-0- a - b - R and a -0- then which of the following is always true-A- roots are real- distinct and have same signB- roots are real and repeatedC- roots are not realD- roots are real- distinct and have opposite sign

The correct option is D roots are real, nbsp;distinct and have opposite signax^2+bx 4=0, a,b∈ℝ and a gt;0 nbsp; D=b^2+16a gt;0 (∵ a gt;0) Real and ...

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Standard XIII

Mathematics

Nature of Roots

For the quadr...

Question

For the quadratic equation $a x^{2} + b x - 4 = 0, a, b \in R$ and $a > 0,$ then which of the following is always true?

roots are real, distinct and have same sign

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roots are real, distinct and have opposite sign

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roots are not real

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roots are real and repeated

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Solution

The correct option is B roots are real, distinct and have opposite sign
$a x^{2} + b x - 4 = 0, a, b \in R$ and $a > 0$

$D = b^{2} + 16 a > 0 (∵ a > 0)$
Real and distinct roots
Product of roots
$= - \frac{4}{a} < 0$
Roots are opposite in sign.

Hence, the roots are real, distinct and have opposite sign.

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Nature of Roots

Standard XIII Mathematics

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For the quadratic equation ax2+bx−4=0, a,b∈R and a>0, then which of the following is always true?

The correct option is B roots are real, distinct and have opposite signax2+bx−4=0, a,b∈R and a>0 D=b2+16a>0 (∵a>0) Real and distinct roots Product of roots =−4a<0 Roots are opposite in sign. Hence, the roots are real, distinct and have opposite sign.

For the quadratic equation $a x^{2} + b x - 4 = 0, a, b \in R$ and $a > 0,$ then which of the following is always true?