If both the roots of ax 2- bx - c -0 are real- positive and distinct- then where - b 2-4 ax A- -0- ab -0- bc -0B- -0- ab 0- ab 0D- -0- ab 0

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Graph of Quadratic Expression

If both the r...

Question

If both the roots of $a x^{2} + b x + c = 0$ are real, positive and distinct, then
(where $Δ = b^{2} - 4 a x$ )

Δ > 0, a b < 0, a c > 0

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Δ > 0, a b < 0, a c < 0

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Δ > 0, a b > 0, b c > 0

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Δ > 0, a b < 0, b c > 0

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Solution

The correct option is A $Δ > 0, a b < 0, a c > 0$
$∵$ Roots are real, positive and distinct
$∴ Δ > 0$ and graph of $f (x) = a x^{2} + b x + c$ will cut the $x -$ axis in positive direction.

Case $1 :$ $a < 0$
Here, $c < 0$ because curve intersects with $y -$ axis in $- v e$ region
$- \frac{b}{2 a} > 0 \Rightarrow b > 0 (∵ x coordinate of vertex is + v e$
$\Rightarrow a b < 0, a c > 0$

Case $2 : a > 0$
Here, $c > 0$ because curve intersects with $y -$ axis in $+ v e$ region.
$- \frac{b}{2 a} > 0 \Rightarrow b < 0 (∵ x coordinate of vertex is + v e$
$\Rightarrow a b < 0, a c > 0$

Alternate solution:
For real, positive and distinct roots.
$Δ > 0$ , sum of roots $> 0$ , product of root $> 0$
$\Rightarrow Δ > 0, - \frac{b}{a} > 0, \frac{c}{a} > 0 \Rightarrow Δ > 0, - \frac{a b}{a^{2}} > 0, \frac{a c}{a^{2}} > 0 \Rightarrow Δ > 0, a b < 0, a c > 0$

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If both the roots of ax2+bx+c=0 are real, positive and distinct, then (where Δ=b2−4ax)

If both the roots of $a x^{2} + b x + c = 0$ are real, positive and distinct, then
(where $Δ = b^{2} - 4 a x$ )