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Discriminant

If a < 0 and ...

Question

If a < 0 and D < 0, what can be inferred about $f (x)$ = $a x^{2} + b x + c$ ?

A

f(x) = is always positive

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B

f(x) is always negative

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C

f(x) = 0 always

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D

cannot determine nature of f(x)

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Solution

The correct option is B
f(x) is always negative

When a < 0,the graph of f(x) = $a x^{2} + b x + c$ will be in the shape of inverted bowl.

Also D < 0 indicates that f(x) does not cut x-axis

So, the graph of f(x) will look as

From graph, we can see that f(x) is always negative

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Similar questions

If a < 0 and D < 0, what can be inferred about $f (x)$ = $a x^{2} + b x + c$ ?

Q. Statement 1 : If

f (x) = a x^{2} + b x + c

a > 0, c < 0

b \in R

, then roots of

f (x) = 0

must be real and distinct .
Statement 2 : If

f (x) = a x^{2} + b x + c,

a > 0, b \in R, b \neq 0

and the roots of

f (x) = 0

are real and distinct, then

c

is necessarily negative real number .

k \in R

lies between the roots of

a x^{2} + b x + c = 0, a < 0

f (x) = a x^{2} + b x + c = 0

f (k)

&

f (x) = a x^{2} + b x + c

g (x) = A x^{2} + b x + λ,

a \neq 0, A \neq 0, a, b, c, A, λ \in R

f (x) = 0

g (x) = 0

are imaginary, then which of the following may be correct?

Q. For a quadratic expression

y = f (x) = a x^{2} + b x + c,

f (α) = 0

and for the rest of values of

x, f (x) < 0.

a

D

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