Question

Solve for \[x\]: \[\dfrac{{14}}{{x + 3}} - 1 = \dfrac{5}{{x - 1}}\]; \[x \ne - 3, - 1\]

Answer 1

Hint: Here we are asked to find the value of the unknown variable \[x\] from the given expression. First, we will make the given expression into a simple equation by doing some modification and simplification. If we get a quadratic equation, we can use the formula to find the roots of that equation.

Formula used: The formula that we need to know before solving the problem:
Let \[a{x^2} + bx + c = 0\] be a quadratic equation then the roots of this equation are given by \[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].

Complete step by step answer:
It is given that \[\dfrac{{14}}{{x + 3}} - 1 = \dfrac{5}{{x - 1}}\] we aim to find the value of the unknown variable \[x\].
Let us first make the given expression into a simple equation by doing some simplifications.
Consider the given expression \[x = \dfrac{{7 \pm \sqrt {49 - 104} }}{2}\]
Let us lake LCM on the left-hand side and simplify it.
\[\dfrac{{14 - x - 3}}{{x + 3}} = \dfrac{5}{{x - 1}}\]
\[\dfrac{{11 - x}}{{x + 3}} = \dfrac{5}{{x - 1}}\]
Now let us cross multiply the denominators on both sides.
\[\left( {11 - x} \right)\left( {x - 1} \right) = 5\left( {x + 3} \right)\]
On simplifying the above expression, we get
\[11x - 11 - {x^2} + x = 5x + 15\]
On further simplification we get
\[12x - 11 - {x^2} = 5x + 15\]
\[{x^2} - 7x + 26 = 0\]
By using the formula of roots of a quadratic equation \[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] we get
\[x = \dfrac{{ - \left( { - 7} \right) \pm \sqrt {{{\left( { - 7} \right)}^2} - 4\left( 1 \right)\left( {26} \right)} }}{{2\left( 1 \right)}}\]
On simplifying the above expression, we get
\[x = \dfrac{{7 \pm \sqrt {49 - 104} }}{2}\]
\[x = \dfrac{{7 \pm \sqrt { - 55} }}{2}\]
\[ \Rightarrow x = \dfrac{{7 + i\sqrt {55} }}{2}\] and \[x = \dfrac{{7 - i\sqrt {55} }}{2}\]
Thus, we have found the values of the unknown variable \[x\] which are complex roots \[x = \dfrac{{7 + i\sqrt {55} }}{2}\] and \[x = \dfrac{{7 - i\sqrt {55} }}{2}\] whose real part is \[\dfrac{7}{2}\] and the imaginary part is \[ \pm \dfrac{{\sqrt {55} }}{2}\].

Note:
In the above problem, it is given that \[x \ne - 3, - 1\] because if the value of the unknown variable \[x\] is equal to \[ - 3\] and \[ - 1\] the denominator becomes zero. We know that anything divided by zero is indefinite so \[x\] cannot be equal to \[ - 3\] and \[ - 1\]. Also, we got that the roots of the given expression are complex since we got a negative number inside the square root.