Question

The value of p for which both the roots of the equation $4{x^2} - 20px + (25{p^2} + 15p - 66) = 0$ are less than 2, lies in
A. $\left( {\dfrac{4}{5},2} \right)$
B. $\left( {2,\infty } \right)$
C. $\left( { - 1,\dfrac{4}{5}} \right)$
D. $\left( { - \infty , - 1} \right)$

Answer 1

Hint: Check the concavity of the given polynomial and form an inequality. Use the fact that the sum of roots is less than 4. Also use the fact that the determinant must be greater than or equal to 0. Solve the three inequalities to get the final answer.

Formula used: Given a quadratic polynomial \[a{x^2} + bx + c\], the sum of roots of the polynomial is $ - \dfrac{b}{a}$ and the discriminant is ${b^2} - 4ac$

Complete step-by-step solution:
The coefficient of ${x^2}$ in the given equation is 4 which is greater than 0. Therefore, the graph of the polynomial $4{x^2} - 20px + (25{p^2} + 15 - 66)$ is concave upwards. Since both the roots are less than 2, $f(2) > 0$. Let the roots be $m,n$.

(Self-made diagram)
Since $f(2) > 0$
$4{(2)^2} - 20p(2) + (25{p^2} + 15p - 66) > 0$
$16 - 40p + 25{p^2} + 15p - 66 > 0$
$25{p^2} - 25p - 50 > 0$
Simplifying further, we get
${p^2} - p - 2 > 0$
${p^2} - 2p + p - 2 > 0$
$(p + 1)(p - 2) > 0$
$p \in ( - \infty , - 1) \cup (2,\infty )$
Since the roots of $4{x^2} - 20px + (25{p^2} + 15p - 66) = 0$ are real, the discriminant must be greater than or equal to 0.
${( - 20p)^2} - 4(4)(25{p^2} + 15p - 66) \geqslant 0$
$400{p^2} - 400{p^2} - 240p + 1056 \geqslant 0$
$240p \leqslant 1056$
Simplifying further, we get
$p \leqslant \dfrac{{22}}{5}$
\[p \in ( - \infty ,4.4]\]
Since both the roots are less than, the sum of roots must be lesser than 4.
$ - \dfrac{{ - 20p}}{4} < 4$
$5p < 4$
Simplifying further, we get
$p < \dfrac{4}{5}$
$p \in \left( { - \infty ,\dfrac{4}{5}} \right)$
Taking the intersection of $( - \infty , - 1) \cup (2,\infty )$, \[( - \infty ,4.4]\] and $\left( { - \infty ,\dfrac{4}{5}} \right)$, we get $p \in ( - \infty , - 1)$
Therefore, the correct answer is option D. $( - \infty , - 1)$

Note: Given a quadratic polynomial \[a{x^2} + bx + c\], the concavity of the graph of the polynomial depends on the sign of $a$; If $a > 0$, the graph will be concave upwards; If $a < 0$ the graph will be concave downwards.