The sum of all the value of m for which the roots x1 and x2 of the quadratic equation x2-2 m x+m=0 satisfy the condition x13+x23=x12+x22, is

Question

The sum of all the value of m for which the roots x1 and x2 of the quadratic equation x2-2 m x+m=0 satisfy the condition x13+x23=x12+x22, is (A) (3/4) (B) 1 (C) (9/4) (D) (5/4)

Tardigrade Admin · Accepted Answer

x1+x2=2 m ; x1 x2=m (x1+x2)3-3 x1 x2(x1+x2)=(x1+x2)2-2 x1 x2 8 m3-3 m(2 m)=4 m2-2 m 8 m3-10 m2+2 m=0 ⇒ m1+m2+m2=(10/8)=(5/4)

Q. The sum of all the value of $m$ for which the roots $x_{1}$ and $x_{2}$ of the quadratic equation $x^{2} - 2 m x + m = 0$ satisfy the condition $x_{1}^{3} + x_{2}^{3} = x_{1}^{2} + x_{2}^{2}$ , is

$\frac{3}{4}$

1

$\frac{9}{4}$

$\frac{5}{4}$

$x_{1} + x_{2} = 2 m; x_{1} x_{2} = m$
$(x_{1} + x_{2})^{3} - 3 x_{1} x_{2} (x_{1} + x_{2}) = (x_{1} + x_{2})^{2} - 2 x_{1} x_{2}$
$8 m^{3} - 3 m (2 m) = 4 m^{2} - 2 m$
$8 m^{3} - 10 m^{2} + 2 m = 0 \Rightarrow m_{1} + m_{2} + m_{2} = \frac{10}{8} = \frac{5}{4}$

Q. The sum of all the value of m for which the roots x1​ and x2​ of the quadratic equation x2−2mx+m=0 satisfy the condition x13​+x23​=x12​+x22​, is

43​

1

49​

45​

x1​+x2​=2m;x1​x2​=m (x1​+x2​)3−3x1​x2​(x1​+x2​)=(x1​+x2​)2−2x1​x2​ 8m3−3m(2m)=4m2−2m 8m3−10m2+2m=0⇒m1​+m2​+m2​=810​=45​

Q. The sum of all the value of $m$ for which the roots $x_{1}$ and $x_{2}$ of the quadratic equation $x^{2} - 2 m x + m = 0$ satisfy the condition $x_{1}^{3} + x_{2}^{3} = x_{1}^{2} + x_{2}^{2}$ , is

$\frac{3}{4}$

$\frac{9}{4}$

$\frac{5}{4}$

$x_{1} + x_{2} = 2 m; x_{1} x_{2} = m$
$(x_{1} + x_{2})^{3} - 3 x_{1} x_{2} (x_{1} + x_{2}) = (x_{1} + x_{2})^{2} - 2 x_{1} x_{2}$
$8 m^{3} - 3 m (2 m) = 4 m^{2} - 2 m$
$8 m^{3} - 10 m^{2} + 2 m = 0 \Rightarrow m_{1} + m_{2} + m_{2} = \frac{10}{8} = \frac{5}{4}$