Question

$35x {}^{2} - 31x - 40$
​

Answer 1

$\large p(x)=35x^2-31x-40 \\ \\ \\ a=35\ \ \ ; \ \ \ b=-31\ \ \ ; \ \ \ c=-40$

$\textsf{We have to split -31 . Let p+q=-31\ \ \ \rightarrow\ \ \ (1) } \\ \\ \\ \textsf{Thus,}\\ \\ pq=ac\ \ \ \ \ \textsf{[Always true]} \\ \\ pq=35 \times -40 \\ \\ pq=-1400\ \ \ \longrightarrow\ \ \ (2)$

$\textsf{Squaring (1) ,} \\ \\ \\ (p+q)^2=(-31)^2 \\ \\ p^2+q^2+2pq=961 \\ \\ \\ \textsf{Now subtract four times (2) from both sides.} \\ \\ \\ p^2+q^2+2pq-4pq=961-(4 \times -1400) \\ \\ p^2+q^2-2pq=961-(-5600) \\ \\ (p-q)^2=961+5600 \\ \\ (p-q)^2=6561 \\ \\ p-q=\sqrt{6561} \\ \\ p-q=\pm 81$

$\textsf{Taking \ p-q=81\ \ \ \rightarrow\ \ \ (3) } \\ \\ \\ (1)+(3) \\ \\ (p+q)-(p-q)=-31+81 \\ \\ 2p=50 \\ \\ p=25 \\ \\ \\ (1)-(3) \\ \\ (p+q)-(p-q)=-31-81 \\ \\ 2q=-112 \\ \\ q=-56$

$\textsf{Therefore, } \\ \\ \\ 35x^2-31x-40 \\ \\ 35x^2+25x-56x-40 \\ \\ 5x(7x+5)-8(7x+5) \\ \\ (7x+5)(5x-8)$

$\displaystyle\textsf{Hence factorised and the roots are \ \frac{-5}{7} \ and \ \frac{8}{5} }\ .$