Discuss the nature of the roots of the equation 4ax2-3bx-2c-0- where a- b- c- R and are connected by the relation a-b-c-0-

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Standard XII

Mathematics

Nature of Roots

Discuss the n...

Question

Discuss the nature of the roots of the equation $4 a x^{2} + 3 b x + 2 c = 0$ , where $a, b, c \in R$ and are connected by the relation $a + b + c = 0$ .

All roots are imaginary.

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All roots are real.

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They can be real or imaginary depends on the value of

a, b, c

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None of these

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Solution

The correct option is D All roots are real.
$Δ = 9 b^{2} - 32 a c = 9 b^{2} + 32 a (a + b) = 9 b^{2} - 32 a b = 9 b^{2} + 32 a^{2} = b^{2} + 8 (b^{2} + 4 a b + 4 a^{2}) = b^{2} + 8 {(b + 2 a)}^{2} = + i v e$
Roots are real

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Similar questions

Q. (a) Discuss the nature of the roots of the equations

4 a x^{2} + 3 b x + 2 c = 0

where

a, b, c ϵ R

and are connected by the relation

a + b + c = 0

.
(b) If the roots of the equation

[a^{2} + 291 - b)] x^{2} + 2 a (1 + b) x + 2 b (b - 1) + a^{2} = 0

be equal, then prove that

a^{2} = 4 b

If a,b,cϵ R and 1 is a root of the equation a $x^{2}$ + bx + c = 0 then the equation 4 a $x^{2}$ + 3 bx + 2c = 0, c ≠ 0 has roots which are :

Q. Assertion :If equation

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

and

x^{2} - 3 x + 4 = 0

have exactly one root common, then at least one of

a, b, c

is imaginary. Reason: If

a, b, c

are not all real, then equation

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

can have one real root and one one root imaginary.

Q. (a) If the roots

α, β

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

be real, then establish between the coefficients under the following conditions:
(i) Roots are equal and opposite.
(ii) Roots are of opposite signs.
(iii) Roots are both

- i v e

(iv) Roots are both

+ i v e

.
(b) Let

a > 0, b > 0

, then both roots of the equation

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

(i) are real and negative
(ii) have negative real parts.
(iii) none of these.
(c) If the roots of the equation

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

be imaginary. then for all real values of x, the expression

3 b^{2} x^{2} + 6 b c x + 2 c^{2}

is
(a) less than

- 4 a b

(b) greater then

4 a b

4 a b

(d) greater than

- 4 a b

Q. In the equation

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

it is given that

D = (b^{2} - 4 a c) > 0 .

Then, the roots of the equation are
(a) real and equal
(b) real and unequal
(c) imaginary
(d) none of these

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Discuss the nature of the roots of the equation 4ax2+3bx+2c=0, where a,b,c∈R and are connected by the relation a+b+c=0.

The correct option is D All roots are real.Δ=9b2−32ac=9b2+32a(a+b)=9b2−32ab=9b2+32a2=b2+8(b2+4ab+4a2)=b2+8(b+2a)2=+iveRoots are real

Discuss the nature of the roots of the equation $4 a x^{2} + 3 b x + 2 c = 0$ , where $a, b, c \in R$ and are connected by the relation $a + b + c = 0$ .

The correct option is D All roots are real.
$Δ = 9 b^{2} - 32 a c = 9 b^{2} + 32 a (a + b) = 9 b^{2} - 32 a b = 9 b^{2} + 32 a^{2} = b^{2} + 8 (b^{2} + 4 a b + 4 a^{2}) = b^{2} + 8 {(b + 2 a)}^{2} = + i v e$
Roots are real