First slide

Theory of equations

Question

If $f (x) = {ax}^{2} + bx + c$ , where $a \neq 0, b, c \in R$ , then which of the following conditions implies that f(x) has real roots?

Moderate

A

a + b + c = 0
B

a and c are of opposite signs
C

$4 ac - b^{2} < 0$
D

a and b are of opposite signs

Solution

(1) If $f (1) = 0 \Rightarrow a + b + c = 0$
So, roots are real.
(2) $D = b^{2} - 4 ac$
If $ac < 0 \Rightarrow D > 0$
So, roots are real.
(3) $4 ac - b^{2} < 0 \Rightarrow b^{2} - 4 ac > 0$
So, roots are real.
(4) This is not the sufficient condition.

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