Question

If one root of \[5{x^2} + 13x + k = 0\] is the reciprocal of the other, then find the value of k.

Answer 1

Hint:
Given equation is a quadratic equation. According to the given condition if one root is reciprocal or other then their product will be 1. So first we will find the roots and then we will use this hint. Find the roots as
\[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]

Complete step by step solution:
Given equation is \[5{x^2} + 13x + k = 0\] of the form \[a{x^2} + bx + c = 0\].
Thus a=5 ,b=13 and c=k.
Now let’s find the roots
\[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
Putting the values,
\[ \Rightarrow \dfrac{{ - 13 \pm \sqrt {{{\left( { - 13} \right)}^2} - 4 \times 5 \times k} }}{{2 \times 5}}\]
\[ \Rightarrow \dfrac{{ - 13 \pm \sqrt {169 - 20k} }}{{10}}\]
Now the two roots are \[ \Rightarrow \dfrac{{ - 13 + \sqrt {169 - 20k} }}{{10}}\] and \[\dfrac{{ - 13 - \sqrt {169 - 20k} }}{{10}}\]
Given that the product of one of them and reciprocal of other. So,
\[\dfrac{{ - 13 + \sqrt {169 - 20k} }}{{10}} = \dfrac{1}{{\dfrac{{ - 13 - \sqrt {169 - 20k} }}{{10}}}}\]
Now we will solve above equation
\[ \Rightarrow \dfrac{{ - 13 + \sqrt {169 - 20k} }}{{10}} = \dfrac{{10}}{{ - 13 - \sqrt {169 - 20k} }}\]
Now cross multiply
\[ \Rightarrow \dfrac{{ - 13 + \sqrt {169 - 20k} }}{{10}} \times \dfrac{{ - 13 - \sqrt {169 - 20k} }}{{10}} = 1\]
In numerator we observe that it is \[\left( {a + b} \right)\left( {a - b} \right)\] so we can use the identity \[\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\]
\[ \Rightarrow \dfrac{{{{\left( { - 13} \right)}^2} - {{\left( {\sqrt {169 - 20k} } \right)}^2}}}{{100}} = 1\]
\[ \Rightarrow \dfrac{{169 - \left( {169 - 20k} \right)}}{{100}} = 1\]
\[ \Rightarrow 169 - 169 + 20k = 100\]
169 is cancelled,
\[ \Rightarrow 20k = 100\]
Divide 100 by 20 we get,
\[ \Rightarrow k = \dfrac{{100}}{{20}}\]
\[ \Rightarrow k = 5\]

Thus value of k is 5.

Note:
Students first we will find the roots and then we will use the condition given. But students may get confused in the word reciprocal or they might forget to change the sign that is one root has plus sign \[\dfrac{{ - b + \sqrt {{b^2} - 4ac} }}{{2a}}\] and other is having minus sign \[\dfrac{{ - b - \sqrt {{b^2} - 4ac} }}{{2a}}\].