Let $x_{1}, x_{2}$ be the roots of $a x^{2} + b x + c = 0$ and $x_{1} . x_{2} < 0$ , $x_{1} + x_{2}$ is non - zero. Roots of $x_{1} (x - x_{2})^{2} + x_{2} (x - x_{1})^{2} = 0$ are

Real and of opposite signs

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

non real

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Negative

No worries! We‘ve got your back. Try BYJU‘S free classes today!

positive

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A
Real and of opposite signs

Given $x_{1} (x - x_{2})^{2} + x_{2} (x - x_{1})^{2} = 0$
$x_{1} (x^{2} + x_{2}^{2} - 2 x x_{2}) + x_{2} (x^{2} + x_{1}^{2} - 2 x x_{1}) = 0$
$\Rightarrow x^{2} (x_{1} + x_{2}) - 4 x x_{1} x_{2} + x_{1} x_{2} (x_{1} + x_{2}) = 0 Δ = 16 x_{1}^{2} x_{2}^{2} - 4 x_{1} x_{2} (x_{1} + x_{2})^{2} > 0 (∵ x_{1} x_{2} < 0 g i v e n)$
$∴$ Real roots
Also, product of roots $= x_{1} x_{2} < 0$
$∴$ Roots are of opposite sign.

Suggest Corrections

Similar questions

Let $x_{1}, x_{2}$ be the roots of $a x^{2} + b x + c = 0$ and $x_{1} . x_{2} < 0$ , $x_{1} + x_{2}$ is non - zero. Roots of $x_{1} (x - x_{2})^{2} + x_{2} (x - x_{1})^{2} = 0$ are

x_{1}

and

x_{2}

are the real roots of

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

and

x_{1} x_{2} < 0

. The roots of

x_{1} (x - x_{2})^{2} + x_{2} (x - x_{1})^{2} = 0

are

Q. If

x_{1}

and

x_{2}

are the roots of

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

and they are of opposite sign, then the roots of

x_{1} (x - x_{2})^{2} + x_{2} (x - x_{1})^{2} = 0

are

If $x_{1}, x_{2}$ are the roots of $a x^{2}$ + bx + c = 0 and $x_{1} + d, x_{2} + d$ are the roots of $p x^{2}$ + qx + r = 0, d $\neq$ 0 then

Q. Let

x_{1}, x_{2}, x_{3} b e 3

roots of the cubic

x^{3} - x - 1 = 0

then the expression

x_{1} {(x_{2} - x_{3})}^{2} + x_{2} {(x_{3} - x_{1})}^{2} + x_{3} {(x_{1} - x_{2})}^{2}

equal a rational number. Find the absolute value of the number.

Join BYJU'S Learning Program

Let x1, x2 be the roots of ax2+bx+c=0 and x1.x2<0, x1+x2 is non - zero. Roots of x1(x−x2)2+x2(x−x1)2=0 are

The correct option is A Real and of opposite signs Given x1(x−x2)2+x2(x−x1)2=0 x1(x2+x22−2xx2)+x2(x2+x21−2xx1)=0 ⇒ x2(x1+x2)−4x x1 x2+x1 x2(x1+x2)=0Δ=16 x21x22−4x1x2(x1+x2)2>0 (∵ x1x2<0 given) ∴ Real roots Also, product of roots =x1x2<0 ∴ Roots are of opposite sign.

Let $x_{1}, x_{2}$ be the roots of $a x^{2} + b x + c = 0$ and $x_{1} . x_{2} < 0$ , $x_{1} + x_{2}$ is non - zero. Roots of $x_{1} (x - x_{2})^{2} + x_{2} (x - x_{1})^{2} = 0$ are