If a - 0 and the equation ax2 - bx - c - 0 has two real roots - and - such that - - - - 1- - - - 1- then which of the following inequalities holds true-

The correct options are A a + b + c ≥ 0 B a b+c≥ 0 C a + | b | + c ≥ 0 D a c ≥ 0 y=ax^2+bx+c since both the roots lie between th ...

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Byju's Answer

Standard XII

Mathematics

Inequalities Involving Modulus Function

If a > 0 and ...

Question

If a > 0 and the equation $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ such that $| α | \leq 1, | β | \leq 1$ , then which of the following inequalities holds true?

a + b + c \geq 0

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a - b + c \geq 0

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a + | b | + c \geq 0

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a - c \geq 0

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Solution

The correct options are
A $a + b + c \geq 0$
B $a - b + c \geq 0$
C $a + | b | + c \geq 0$
D $a - c \geq 0$
$y = a x^{2} + b x + c$
since both the roots lie between the interval of [-1,1]
$y (- 1) \geq 0$ and $y (1) \geq 0$
Therefore
$a - b + c \geq 0$ and $a + b + c \geq 0$
Therefore
$a + | b | + c \geq 0$
Also $α β \leq | α β | \leq | α | | β | \leq 1$
Therefore
$\frac{c}{a} \leq 1$
$c \leq a$
$c - a \leq 0$
hence all options are correct

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If a > 0 and the equation ax2+bx+c=0 has two real roots α and β such that |α|≤1,|β|≤1, then which of the following inequalities holds true?

If a > 0 and the equation $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ such that $| α | \leq 1, | β | \leq 1$ , then which of the following inequalities holds true?