If the roots of the quadratic equation a x -12-2 b x -2- c x -1-4-0 are imaginary- where a - b - c - R and b -2- thenA- a - c -4C- 3 c-8 b-9 a-4D- 3 c-8 b-9 a-4

The correct options are A a+c lt; 4 C 3c+8b 9a gt;4Let, f(x)=a(x 1)^2+2b(x 2)+c(x 1)+4 f(1)= 2b+4⇒ f(1) lt;0 (∵ b gt;2) ∴ f(x) lt;0 ∀ x∈ℝ f(2)=a+c ...

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Byju's Answer

Standard XII

Mathematics

Relation between Roots and Coefficients for Quadratic

If the roots ...

Question

If the roots of the quadratic equation $a (x - 1)^{2} + 2 b (x - 2) + c (x - 1) + 4 = 0$ are imaginary, where $a, b, c \in R$ and $b > 2$ , then

a + c < - 4

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a + c > - 4

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3 c + 8 b - 9 a > 4

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3 c + 8 b - 9 a < 4

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Solution

The correct options are
A $a + c < - 4$
C $3 c + 8 b - 9 a > 4$
Let, $f (x) = a (x - 1)^{2} + 2 b (x - 2) + c (x - 1) + 4$
$f (1) = - 2 b + 4 \Rightarrow f (1) < 0 (∵ b > 2)$
$∴ f (x) < 0 \forall x \in R$

$f (2) = a + c + 4 < 0 \Rightarrow a + c < - 4$
$f (- 2) = 9 a - 8 b - 3 c + 4 < 0$
$\Rightarrow 3 c - 9 a + 8 b > 4$

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If the roots of the quadratic equation a(x−1)2+2b(x−2)+c(x−1)+4=0 are imaginary, where a,b,c∈R and b>2, then

The correct options are A a+c<−4 C 3c+8b−9a>4Let, f(x)=a(x−1)2+2b(x−2)+c(x−1)+4 f(1)=−2b+4⇒ f(1)<0 (∵b>2) ∴f(x)<0 ∀ x∈R f(2)=a+c+4<0⇒a+c<−4 f(−2)=9a−8b−3c+4<0 ⇒ 3c−9a+8b>4

If the roots of the quadratic equation $a (x - 1)^{2} + 2 b (x - 2) + c (x - 1) + 4 = 0$ are imaginary, where $a, b, c \in R$ and $b > 2$ , then