Question

Solve the following equation for \[x\]:
$\dfrac{{16}}{x} - 1 = \dfrac{{15}}{{x + 1}},x \ne 0,x \ne - 1$.

Answer 1

Hint: First of all, we will simplify the LHS to get fraction only and then cross multiply it with RHS. Then simplifying by multiplying required values will lead us to a quadratic equation and solving that keeping in mind that x cannot be equal to 0 and 1 would give us the answer.

Complete step-by-step answer:
We have with us the following equation in terms of x:-
$\dfrac{{16}}{x} - 1 = \dfrac{{15}}{{x + 1}},x \ne 0,x \ne - 1$
Taking the LCM of the LHS and keeping the condition aside for once. We will have:-
$\dfrac{{16 - x}}{x} = \dfrac{{15}}{{x + 1}}$
Now, cross multiplying the terms will lead us to:-
$(16 - x)(x + 1) = 15x$
Now simplifying it by opening the bracket on LHS. We will get:-
$ \Rightarrow 16x - {x^2} + 16 - x = 15x$
Now further simplifying it by clubbing the like terms together, we will get:-
$ \Rightarrow - {x^2} + 16 = 0$
Rearranging the terms to get the following expression:-
$ \Rightarrow {x^2} = 16$
This leads us to the following possible values of x:-
$ \Rightarrow x = \pm 4$
Hence, x can take the value either 4 or -4.

Note: The students must notice that cross multiplication is possible only because we have the conditions that it is non zero or not equal to -1. Because if that would have been possible, we are indirectly multiplying both sides by zero, which is not allowed and the denominator can never be zero as well.
Fun Fact:- The function $f(x) = a{x^2} + bx + c$ is a quadratic function. The graph of any quadratic function has the same general shape, which is called a parabola. The location and size of the parabola, and how it opens, depend on the values of a, b, and c. As shown in Figure 1, if a > 0, the parabola has a minimum point and opens upward. If a < 0, the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex.
One property of this form is that it yields one valid root when a = 0, while the other root contains division by zero, because when a = 0, the quadratic equation becomes a linear equation, which has one root. By contrast, in this case, the more common formula has a division by zero for one root and an indeterminate form 0/0 for the other root. On the other hand, when c = 0, the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form 0/0.