if the difference of the roots of equation x2 +kx+12=0 is 1,then find the positive value of k

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if the difference of the roots of equation x2 +kx+12=0 is 1,then
find the positive value of k

Priyanka Kedia · Accepted Answer

The given equation is as,

$x^{2} + k x + 12 = 0 ...... (1)$

Let the roots of the given equation is a and b. Given, the difference of the roots is 1.

Therefore,

$a - b = 1 ...... (2)$

We know that,

$S u m o f r o o t s = - \frac{c o e f f i c i e n t o f x}{c o e f f i c i e n t o f x^{2}} \Rightarrow a + b = - \frac{k}{1} \Rightarrow a + b = - k ...... (3)$

And,

$\Pr o d u c t o f r o o t s = \frac{c o n s \tan t t e r m}{c o e f f i c i e n t o f x^{2}} \Rightarrow a b = \frac{12}{1} \Rightarrow a b = 12 ...... (3)$

Firstly we will solve equation (1) and (2). So, adding equation (1) and (2), we get,

$2 a = 1 - k \Rightarrow a = \frac{1 - k}{2}$

Substitute value of a in equation (1), we get,

$\frac{1 - k}{2} - b = 1 \Rightarrow b = \frac{1 - k}{2} - 1 \Rightarrow b = \frac{- k - 1}{2}$

Now, Substitute value of a and b in equation (3), we get,

$(\frac{1 - k}{2}) (\frac{- 1 - k}{2}) = 12 \Rightarrow (1 - k) (- 1 - k) = 48 \Rightarrow - 1 - k + k + k^{2} = 48 \Rightarrow k^{2} = 49 \Rightarrow k = \pm 7$

Thus, the positive value of k is 7.