Question

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Answer 1

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Question : If the sum of the roots of a the equation x² - (k + 6)x + 2(2k + 1) = 0 is equal to half of their product, then k = ?

Options:

a) 6                  b)7

c)1                     d)5

Solution:

We know that the if a general quadratic equation is ax² + bx + c = 0 has roots α & β, then,

→ Sum of roots, α + β = $-\frac{b}{a}$

→ Product of roots, αβ = $\frac{c}{a}$

In the following quadratic equation,

$\star \ a = 1$

$\star \ b = - ( k + 6)$

$\star \ c = 2(2k + 1)$

Hence,

→ sum of roots, α + β = $-(-\frac{k + 6}{1})$ = $k + 6$

→ product of roots, αβ = $\frac{2(2k + 1)}{1} = 4k + 2$

Now,

According to the question,

$sum \ of \ roots = \frac{1}{2}\ product \ of \ roots$

$\implies \alpha + \beta = \frac{1}{2} \alpha \beta$

$\implies k + 6 = \frac{1}{2}( 4k + 2)$

$\implies 2k + 12 = 4k + 2\\\\\implies 2k = 10\\\\\implies k = 5$

Answer: Option d → k = 5

I hope it helps you