If a x 2 + b x + c ; a , b , c ∈ R has no real zeroes, and if c < 0 , then

Let f ( x ) = a x 2 + b x + c . Since f ( x ) has no real zeroes, either f ( x ) > 0 or f ( x ) < 0 for all x ∈ R . since f ( 0 ) = c < 0 , we get f ( x ) < 0 for all x ∈ R . Therefore, a < 0 as the parabola y = f ( x ) must open downward. Obviously f ( 1 ) , f ( − 1 ) and f ( 2 ) < 0 .

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Mathematics - Quadratic Equation Question with Solution | TestHub

MathematicsQuadratic EquationGraph and Sign of QuadraticMedium2 minQB

MathematicsMediumsingle choice

If $a x^{2} + b x + c; a, b, c \in R$ has no real zeroes, and if $c < 0$ , then

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Let $f (x) = a x^{2} + b x + c$ . Since $f (x)$ has no real zeroes, either $f (x) > 0$ or $f (x) < 0$ for all $x \in R$ .

since $f (0) = c < 0$ , we get $f (x) < 0$ for all $x \in R$ . Therefore, $a < 0$ as the parabola $y = f (x)$ must open downward. Obviously $f (1), f (- 1)$ and $f (2) < 0$ .

Stream:JEESubject:MathematicsTopic:Quadratic EquationSubtopic:Graph and Sign of Quadratic

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