For the quadratic equation ax 2- bx - c -0- a - b - c - R and ac -0- then which of the following is true -A- roots are real- distinct and have same signB- roots are real- distinct and have opposite signC- roots are not realD- roots are real and repeated

The correct option is B roots are real, nbsp;distinct and have opposite signax^2+bx+c=0, a,b,c∈ℝ Δ=b^2 4ac gt;0 (∵ ac lt;0) b^2 4ac gt;b^2 ⇒√(Δ) gt; ...

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Standard XIII

Mathematics

Nature of Roots

For the quadr...

Question

For the quadratic equation $a x^{2} + b x + c = 0, a, b, c \in R$ and $a c < 0,$ then which of the following is 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
$a x^{2} + b x + c = 0, a, b, c \in R$
$Δ = b^{2} - 4 a c > 0 (∵ a c < 0)$
$b^{2} - 4 a c > b^{2}$
$\Rightarrow \sqrt{Δ} > b$
$x = \frac{- b \pm \sqrt{Δ}}{2 a}$
$- b + \sqrt{Δ} \to + v e$
$- b - \sqrt{Δ} \to - v e$
Hence, 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+c=0, a,b,c∈R and ac<0, then which of the following is true ?

The correct option is B roots are real, distinct and have opposite signax2+bx+c=0, a,b,c∈R Δ=b2−4ac>0 (∵ac<0) b2−4ac>b2 ⇒√Δ>b x=−b±√Δ2a −b+√Δ→+ve −b−√Δ→−ve Hence, roots are real, distinct and have opposite sign

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