Question

the value of p for which both roots of the equation 4x²-20px(25p²+15p-66)=0 are less than 2 lies in​

Answer 1

EXPLANATION.

value of P for which both the equation

$4 {x}^{2} - 20px(25 {p}^{2} + 15p - 66) = 0$

are less than 2

TO FIND VALUE OF P.

according to the question,

given,

First conditions

$4 {x}^{2} - 20px(25 {p}^{2} + 15p - 66) = 0$

Therefore,

$d \geqslant 0$

$( - 20p) {}^{2} - 4 \times 4(25 {p}^{2} + 15p - 66) \geqslant 0$

$16(25 {p}^{2} - 25 {p}^{2} - 15p + 66) \geqslant 0$

$- 15p + 66 \geqslant 0$

$15p - 66 \leqslant 0$

$p \leqslant \frac{66}{15}$

second conditions

$\frac{ - b}{2a} < 2$

$\frac{20p}{8} < 2$

$p < \frac{16}{20}$

$p < \frac{4}{5}$

Third conditions

$f(2) > 0$

$16 - 40p + 25 {p}^{2} + 15p - 66 > 0$

$25 {p}^{2} - 25p - 50 > 0$

${p}^{2} - p - 2 > 0$

$(p - 2)(p + 1) > 0$

$p \in( - \infty \:, - 1) \cup(2 \: , \infty)$