Question

24. If ꭤ , β are the roots of the equation 2x² - 6x + k =0 and ꭤ +3 β =9, find the value of k.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\: \alpha , \beta \: are \: the \: roots \: {2x}^{2} - 6x + k = 0$

such that

$\rm :\longmapsto\: \alpha + 3 \beta = 9 - - - (1)$

Now, we know that

$\boxed{\purple{\tt Sum\ of\ the\ zeroes=\frac{-b}{a}}}$

$\rm :\longmapsto\: \alpha + \beta = - \dfrac{( - 6)}{2}$

$\rm :\longmapsto\: \alpha + \beta =3 - - - - (2)$

On Subtracting equation (2) from equation (1), we get

$\rm :\longmapsto\:2 \beta = 6$

$\rm :\longmapsto\:\beta = 3$

On substituting this value in equation (2), we get

$\rm :\longmapsto\: \alpha + \beta = 3$

$\rm :\longmapsto\: \alpha + 3 = 3$

$\rm :\longmapsto\: \alpha = 0$

$\boxed{\red{\sf Product\ of\ the\ zeroes=\frac{Constant}{coefficient\ of\ x^{2}}}}$

$\rm :\longmapsto\: \alpha \beta = \dfrac{k}{2}$

Now, on substituting the values, we get

$\rm :\longmapsto\: 0 \times 3 = \dfrac{k}{2}$

$\rm :\longmapsto\: 0 = \dfrac{k}{2}$

$\bf\implies \:k = 0$

Additional Information:-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac