Roots of the quadratic equation x2-4x-3- x2-6x-8-0- R will be

(x^2 4x+3)+λ(x^2 6x+8)=0 x^2(1+λ) 2x(2+3λ)+(3+8λ)=0 Discriminant. D=4(2+3λ)^2 4(1+λ)(3+8λ) D=4(λ^2+λ+1) If λ∈ R then D 0 So, root of given quadratic are al ...

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Standard XII

Mathematics

Nature of Roots

Roots of the ...

Question

Roots of the quadratic equation $(x^{2} - 4 x + 3) + λ (x^{2} - 6 x + 8) = 0, λ \in R$ will be

Real only when

λ

is negative

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Always real

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Always imaginary

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Real only when

λ

is positive

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Solution

The correct option is B Always real
$(x^{2} - 4 x + 3) + λ (x^{2} - 6 x + 8) = 0$

$x^{2} (1 + λ) - 2 x (2 + 3 λ) + (3 + 8 λ) = 0$

Discriminant.
$D = 4 (2 + 3 λ)^{2} - 4 (1 + λ) (3 + 8 λ)$

$D = 4 (λ^{2} + λ + 1)$

If $λ \in R$ then $D > 0$

So, root of given quadratic are always real.

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Similar questions

Q. The quadratic equation whose one root is

(3 + 2 \sqrt{3})

is
(a)

x^{2} + 6 x - 3 = 0

(b)

x^{2} - 6 x - 3 = 0

(c)

x^{2} + 6 x + 3 = 0

(d)

x^{2} - 6 x + 3 = 0

Q. Roots of the quadratic equation

(x^{2} - 4 x + 3) + λ (x^{2} - 6 x + 8) = 0, λ \in R

will be

Q. The equation

x^{2} - 6 x + 8 + λ (x^{2} - 4 x + 3) = 0, λ \in R

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Q. Roots of the quadratic equation

(x^{2} - 4 x + 3) + λ (x^{2} - 6 x + 8) = 0, λ \in R

will be

Q. The equation

x^{2} - 6 x + 8 + λ (x^{2} - 4 x + 3) = 0, \forall λ \in R - {- 1}

has

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Roots of the quadratic equation (x2−4x+3)+λ(x2−6x+8)=0,λ∈R will be

The correct option is B Always real(x2−4x+3)+λ(x2−6x+8)=0 x2(1+λ)−2x(2+3λ)+(3+8λ)=0 Discriminant. D=4(2+3λ)2−4(1+λ)(3+8λ) D=4(λ2+λ+1) If λ∈R then D>0 So, root of given quadratic are always real.

Roots of the quadratic equation $(x^{2} - 4 x + 3) + λ (x^{2} - 6 x + 8) = 0, λ \in R$ will be

The correct option is B Always real
$(x^{2} - 4 x + 3) + λ (x^{2} - 6 x + 8) = 0$

$x^{2} (1 + λ) - 2 x (2 + 3 λ) + (3 + 8 λ) = 0$

Discriminant.
$D = 4 (2 + 3 λ)^{2} - 4 (1 + λ) (3 + 8 λ)$

$D = 4 (λ^{2} + λ + 1)$

If $λ \in R$ then $D > 0$

So, root of given quadratic are always real.