pq-Formula 2021-07-24 61 words algebra maths education We want to find a1a_1a1 and a2a_2a2 using only ppp and qqqx2−(a1+a2)⏟px+a1a2⏟q=(x−a1)(x−a2). x^2 - \underbrace{(a_1+a_2)}_{p}x + \underbrace{a_1a_2}_{q} = (x-a_1)(x-a_2). x2−p(a1+a2)x+qa1a2=(x−a1)(x−a2).Withq=a1a2=(m−d)(m+d)=m2−d2, q= a_1a_2= (m-d)(m+d) = m^2 - d^2, q=a1a2=(m−d)(m+d)=m2−d2,we immediately get the p-q formulaa1/2=m±d=m±m2−q=p2±(p2)2−q \begin{aligned} a_{1/2} &= m \pm d \\ &= m \pm \sqrt{m^2 - q}\\ &= \frac{p}{2} \pm \sqrt{\left(\frac{p}{2}\right)^2 - q} \end{aligned} a1/2=m±d=m±m2−q=2p±(2p)2−qSource: 3blue1brown
