EduGoog

Solution of quadratic equations, Relation between roots and coefficients (only real roots to be considered). Practical problem on quadratic equations.

If the ratio of the roots of the equation $x^2 + 6x + k - 5 = 0$ is 3:5, then find the value of k.

Option:
A. $52/15$
B. $52/18$
C. $55/16$
D. $50/16$
Answer: C . $55/16$

Justification:

Suppose the roots of the equation $x^2 - 6x + k + 5 = 0$ be α and β such that α < β.
Here, a = 1, b = - 6, c = k + 5
$α + β = {-b}/a = 6 $ and $ αβ= c/a = {k + 5}/1 = k + 5 $

Since the ration of the roots is 3:5 $α/β = 3/5$
.`. $α/3 = β/5$
Each ration = $ {α + β}/8$
= $6/8 = 3/4$
.`.$α/3 = 3/4 → α = 9/4 $ and $ β/5 = 3/4 → β = 15/4 $
Now, α.β = k + 5
.`. $9/4$.$15/4 = k + 5$
.`. 135 = 16k + 80
.`. 16k = 135 - 80
.`. 16k = 55
.`. $k = 55/16$

Next