Question

If both \[x - 2\] and \[x - \dfrac{1}{2}\]are factors of \[{\text{p}}{{\text{x}}^{\text{2}}}{\text{ + 5x + r}}\] then
A. p= -r
B. p= r= 0
C. p= r= -1
D. p= r

Answer 1

Hint: Here we are given the values of x so we will put these values in the given equation one by one and solve the two obtained equations simultaneously to get the relation between p and r.

Complete step-by-step answer:
We are given with the following equation:
\[{\text{p}}{{\text{x}}^{\text{2}}}{\text{ + 5x + r}} = 0\]
And the roots of this equation are:
\[x - 2 = 0\] and \[x - \dfrac{1}{2} = 0\]
Now we will find the values of x from the equations of roots so,
\[{\text{x}} = 2{\text{ }}\]and \[{\text{x}} = \dfrac{1}{2}\]
Now we will put these values of x one by one in the given equation to get the desired relation.
Therefore, putting \[{\text{x}} = 2{\text{ }}\]in the equation \[{\text{p}}{{\text{x}}^{\text{2}}}{\text{ + 5x + r}} = 0\] we get:-
\[
  {\text{p}}{\left( 2 \right)^{\text{2}}}{\text{ + 5}}\left( 2 \right){\text{ + r}} = 0 \\
  4{\text{p}} + 10 + {\text{r}} = 0 \\
  4{\text{p}} + {\text{r}} = - 10........................\left( 1 \right) \\
 \]
Now putting \[{\text{x}} = \dfrac{1}{2}\] in the equation \[{\text{p}}{{\text{x}}^{\text{2}}}{\text{ + 5x + r}} = 0\] we get:-
\[
  {\text{p}}{\left( {\dfrac{1}{2}} \right)^{\text{2}}}{\text{ + 5}}\left( {\dfrac{1}{2}} \right){\text{ + r}} = 0 \\
  \dfrac{{\text{p}}}{4} + \dfrac{5}{2} + {\text{r}} = 0 \\
  \dfrac{{{\text{p}} + 10 + 4{\text{r}}}}{4} = 0 \\
  {\text{p}} + 10 + 4{\text{r}} = 0 \\
  {\text{p}} + 4{\text{r}} = - 10........................\left( 2 \right) \\
 \]
Since the right hand side of both the equations is equal therefore we can equate equation 1 and equation 2.
Therefore equating equation 1 and equation 2 we get:-
\[
  {\text{4p}} + {\text{r}} = {\text{p}} + 4{\text{r}} \\
  4{\text{p}} - {\text{p}} = 4{\text{r}} - {\text{r}} \\
  3{\text{p}} = 3{\text{r}} \\
  {\text{p}} = {\text{r}} \\
 \]
Hence we got the relation between p and r as :
\[{\text{p}} = {\text{r}}\]
Therefore option D is correct.

Note: The alternative method to solve this question is to make a quadratic equation by multiplying the given roots and then comparing the equation so formed with the given equation to get the desired relation.