Question

Sum of the roots of a quadratic equation is double their product. Find K if equation is x²– 4kx + k+ 3 = 0​

Answer 1

Answer:

k=3

Step-by-step explanation:

Let α and β are the roots of quadratic equation x² - 4kx + k + 3 = 0

∴sum of roots = - coefficient of x²/coefficient of x

α + β = -(-4k)/1 = 4k

Product of roots = constant/coefficient of x²

αβ = (k + 3)/1 = (k + 3)

A/C to question,

sum of roots = 2 × product of roots

(α + β ) = 2αβ

⇒4k = 2(k + 3)

⇒ 4k = 2k + 6

⇒ 4k - 2k = 6

⇒ 2k = 6

⇒ k = 3

Hence, answer is k = 3

Answer 2

$\large{\underline{\sf{\orange{Given-}}}}$

Quadratic Equation - x² - 4kx + k + 3 = 0

$\large{\underline{\sf{\pink{To \: Find-}}}}$

Value of k.

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

Here,

a = 1

b = -4k

c = k + 3

$:\implies\rm{\alpha + \beta = 2 \alpha \beta } \: .... \: (1)$

$:\implies\rm{\alpha + \beta = - \frac{b}{a}}$

$:\implies\rm{\alpha + \beta = - \frac{b}{a}}$

$:\implies\rm{\alpha + \beta = - \frac{ - 4k}{1} = 4k} \: ... \: (2)$

$:\implies\rm{ \alpha \beta = \frac{c}{a} = \frac{k + 3}{1} }$

$:\implies\rm{\alpha \beta = k + 3 }\: ... \: (3)$

From (1), (2) and (3),

$:\implies\rm{4k = 2(k + 3)}$

$:\implies\rm{4k - 2k = 6}$

$:\implies\rm{2k = 6}$

$\boxed{\rm\green{k = 3}}$

Hence,

The Value of k is 3.