[tex]S = 75x^2-76x+77x^2-78x+...+171x^2-172x \\ \\ \displaystyle S =\sum\limits_{k=38}^{86}\Big[(2k-1)x^2-2kx\Big] =\\\\ S = \sum\limits_{k=1}^{49}\Big[\Big(2(k+37)-1\Big)x^2 - 2(k+37)x\Big] \\ \\ S = \sum\limits_{k=1}^{49} \Big[(2k+73)x^2 - (2k+74)x\Big] \\ \\ S = (49\cdot 50+73\cdot 49)x^2 - (49\cdot 50 + 74\cdot 49)x \\ \\ S = 49(50+73)x^2-49(50+74)x[/tex]
[tex]S = 49\cdot 123x^2-49\cdot 124x \\ \\ 49\cdot 123 = a \Rightarrow 49\cdot 124 = a+49 \\ \\ S = ax^2-(a+49)x\\ \\ S = a\Big(x-\dfrac{a+49}{2a}\Big)^2 - \dfrac{(a+49)^2}{4a}\\ \\ S = 6027\Big(x - \dfrac{6076}{12054}\Big)^2 - \dfrac{6076^2}{24108}[/tex]
