If α and β are roots of the quadratic equation x2+4x+3=0, then the equation whose roots are 2α text+ β and α text+2 β is

Question

If α and β are roots of the quadratic equation x2+4x+3=0, then the equation whose roots are 2α text+ β and α text+2 β is (A)  x2-12x+35=0  (B)  x2+12x-33=0  (C)  x2-12x-33=0  (D)  x2+12x+35=0

Tardigrade Admin · Accepted Answer

Given α ,β are the roots of equation x2+4x+3=0 ∴ α +β =-4 and α β =3 Now, 2α +β +α +2β =3(α +β )=-12 and (2α +β )(α +2β )=2α 2+4α β +α β +2β 2 =2(α +β )2+α β =2(-4)2+3=35 Hence, required equation is x2-( textsum of roots) x + (product of roots) = 0 ⇒ x2+12x+35=0

Q. If $α$ and $β$ are roots of the quadratic equation $x^{2} + 4 x + 3 = 0,$ then the equation whose roots are $2 α + β$ and $α +2 β$ is

$x^{2} - 12 x + 35 = 0$

$x^{2} + 12 x - 33 = 0$

$x^{2} - 12 x - 33 = 0$

$x^{2} + 12 x + 35 = 0$

Q. If α and β are roots of the quadratic equation x2+4x+3=0, then the equation whose roots are 2α+β and α+2β is

x2−12x+35=0

x2+12x−33=0

x2−12x−33=0

x2+12x+35=0

Given α,β are the roots of equation x2+4x+3=0 ∴ α+β=−4 and αβ=3 Now, 2α+β+α+2β=3(α+β)=−12 and (2α+β)(α+2β)=2α2+4αβ+αβ+2β2 =2(α+β)2+αβ =2(−4)2+3=35 Hence, required equation is x2−(sum of roots) x + (product of roots) = 0 ⇒ x2+12x+35=0

Q. If $α$ and $β$ are roots of the quadratic equation $x^{2} + 4 x + 3 = 0,$ then the equation whose roots are $2 α + β$ and $α +2 β$ is

$x^{2} - 12 x + 35 = 0$

$x^{2} + 12 x - 33 = 0$

$x^{2} - 12 x - 33 = 0$

$x^{2} + 12 x + 35 = 0$