Question Number 226619 by fantastic2 last updated on 07/Dec/25
$${factorise} \\ $$$${x}^{\mathrm{2}} −{qx}−{p}^{\mathrm{2}} +\mathrm{5}{pq}−\mathrm{6}{q}^{\mathrm{2}} \\ $$
Commented by Frix last updated on 07/Dec/25
$${x}^{\mathrm{2}} −{qx}−{p}^{\mathrm{2}} +\mathrm{5}{pq}−\mathrm{6}{q}^{\mathrm{2}} \\ $$$$\left({x}−\frac{{q}}{\mathrm{2}}\right)^{\mathrm{2}} −{p}^{\mathrm{2}} +\mathrm{5}{pq}−\frac{\mathrm{25}}{\mathrm{4}}{q}^{\mathrm{2}} \\ $$$$\left({x}−\frac{{q}}{\mathrm{2}}\right)^{\mathrm{2}} −\left({p}−\frac{\mathrm{5}}{\mathrm{2}}{q}\right)^{\mathrm{2}} \\ $$$$\left(\left({x}−\frac{{q}}{\mathrm{2}}\right)+\left({p}−\frac{\mathrm{5}}{\mathrm{2}}{q}\right)\right)\left(\left({x}−\frac{{q}}{\mathrm{2}}\right)−\left({p}−\frac{\mathrm{5}}{\mathrm{2}}{q}\right)\right) \\ $$$$\left({x}+{p}−\mathrm{3}{q}\right)\left({x}−{p}+\mathrm{2}{q}\right) \\ $$
Commented by fantastic2 last updated on 08/Dec/25
$${thank}\:{you} \\ $$
Commented by fantastic2 last updated on 08/Dec/25
$${x}^{\mathrm{2}} −{qx}−{p}^{\mathrm{2}} +\mathrm{5}{pq}−\mathrm{6}{q}^{\mathrm{2}} \\ $$$$={x}^{\mathrm{2}} −{qx}−\left({p}^{\mathrm{2}} −\mathrm{5}{pq}+\mathrm{6}{q}^{\mathrm{2}} \right) \\ $$$$={x}^{\mathrm{2}} −{qx}−\left({p}^{\mathrm{2}} −\mathrm{3}{pq}−\mathrm{2}{pq}+\mathrm{6}{q}^{\mathrm{2}} \right) \\ $$$$={x}^{\mathrm{2}} −{qx}−\left({p}−\mathrm{3}{q}\right)\left({p}−\mathrm{2}{q}\right) \\ $$$$={x}^{\mathrm{2}} +{x}\left\{\left({p}−\mathrm{3}{q}\right)−\left({p}−\mathrm{2}{q}\right)\right\}−\left({p}−\mathrm{3}{q}\right)\left({p}−\mathrm{2}{q}\right) \\ $$$$={x}^{\mathrm{2}} +{x}\left({p}−\mathrm{3}{q}\right)−{x}\left({p}−\mathrm{2}{q}\right)−\left({p}−\mathrm{3}{q}\right)\left({p}−\mathrm{2}{q}\right) \\ $$$$={x}\left({x}+{p}−\mathrm{3}{q}\right)−\left({p}−\mathrm{2}{q}\right)\left({x}+{p}−\mathrm{3}{q}\right) \\ $$$$=\left({x}+{p}−\mathrm{3}{q}\right)\left({x}−{p}+\mathrm{2}{q}\right)\checkmark \\ $$