D est sous la forme ${\displaystyle a^{2}-b^{2}}$

${\displaystyle D=9x^{2}-1}$

${\displaystyle =\left(3x\right)^{2}-\left(1\right)^{2}}$

${\displaystyle =\left(3x+1\right)\left(3x-1\right)}$

c) ${\displaystyle E=\left(3x+1\right)^{2}+9x^{2}-1}$

${\displaystyle =\left(9x^{2}+2\times 3x\times 1+1\right)+9x^{2}-1}$

${\displaystyle =9x^{2}+6x+1+9x^{2}-1}$

${\displaystyle =18x^{2}+6x}$

${\displaystyle E=18x^{2}+6x}$

${\displaystyle E=6x\left(3x+1\right)}$

${\displaystyle 6x\left(3x+1\right)=0}$

${\displaystyle {\begin{cases}6x=0\\3x+1=0\end{cases}}}$

${\displaystyle {\begin{cases}x=0\\3x=-1\end{cases}}}$

${\displaystyle {\begin{cases}x=0\\x={\frac {-1}{3}}\end{cases}}}$

${\displaystyle S=\left\{0;{\frac {-1}{3}}\right\}}$

${\displaystyle K\left(x\right)=\left(5x-3\right)^{2}+6\left(5x-3\right)}$

= ${\displaystyle \left(25x^{2}-2\times 5x\times 3+9\right)+6\left(5x-3\right)}$

= ${\displaystyle 25x^{2}-30x+9+30x-18}$

= ${\displaystyle 25x^{2}-9}$

${\displaystyle K\left({\sqrt {2}}\right)=25\left({\sqrt {2}}\right)^{2}-9}$

${\displaystyle K\left({\sqrt {2}}\right)=25\times 2-9}$

${\displaystyle K\left({\sqrt {2}}\right)}$ = 41

${\displaystyle \left(3-{\sqrt {5}}\right)^{2}}$

= ${\displaystyle 3^{2}-2\times 3\times {\sqrt {5}}+{\sqrt {5}}^{2}}$

= ${\displaystyle 9-6\times {\sqrt {5}}+5}$

= ${\displaystyle 14-6\times {\sqrt {5}}}$

${\displaystyle E=\left(x-3\right)^{2}-\left(x-1\right)\left(x-2\right)}$

= ${\displaystyle \left(x^{2}-2\times 3\times x+9\right)-\left(x^{2}-2x-x+2\right)}$

= ${\displaystyle x^{2}-6x+9-x^{2}+2x+x-2}$

= ${\displaystyle -3x+7}$

Il suffit de prendre x = 1000000

${\displaystyle 99997^{2}-99999\times 99998=-3\times 1000000+7}$

${\displaystyle 99997^{2}-99999\times 99998=-2999993}$

${\displaystyle F=\left(4x+1\right)^{2}-\left(4x+1\right)\left(7x-6\right)}$

= ${\displaystyle \left[4x+1\right]\left[\left(4x+1\right)-\left(7x-6\right)\right]}$

${\displaystyle \left(4x+1\right)\left(7-3x\right)=0}$

${\displaystyle {\begin{cases}4x+1=0\\7-3x=0\end{cases}}}$

${\displaystyle {\begin{cases}4x=-1\\-3x=-7\end{cases}}}$

${\displaystyle {\begin{cases}x=-{\frac {1}{4}}\\x={\frac {7}{3}}\end{cases}}}$

${\displaystyle \left(4-{\sqrt {5}}\right)^{2}}$

= ${\displaystyle 4^{2}-2\times 4\times {\sqrt {5}}+{\sqrt {5}}^{2}}$

= ${\displaystyle 16-8{\sqrt {5}}+5}$

= ${\displaystyle 21-8{\sqrt {5}}}$

${\displaystyle \left(4x-3\right)^{2}}$

= ${\displaystyle 16x^{2}-2\times 4x\times 3+9}$

= ${\displaystyle 16x^{2}-24x+9}$

${\displaystyle F=\left(4x-3\right)^{2}-\left(x+3\right)\left(3-9x\right)}$

=${\displaystyle 16x^{2}-24x+9-\left(x+3\right)\left(3-9x\right)}$

=${\displaystyle 16x^{2}-24x+9-\left(3x-9x^{2}+9-27x\right)}$

=${\displaystyle 16x^{2}-24x+9-3x+9x^{2}-9+27x}$

=${\displaystyle 25x^{2}}$

=${\displaystyle \left(5x\right)^{2}}$

${\displaystyle F=\left(5x\right)^{2}}$

${\displaystyle \left(5x\right)^{2}=125}$

${\displaystyle \left(5x\right)=\pm {\sqrt {125}}}$

${\displaystyle \left(5x\right)=\pm {\sqrt {25\times 5}}}$

${\displaystyle 5x=\pm 5{\sqrt {5}}}$

${\displaystyle x=\pm {\sqrt {5}}}$

${\displaystyle E=(x-1)^{2}-4}$

${\displaystyle E=(-1)^{2}-4}$

${\displaystyle E=1-4}$

${\displaystyle E=-3}$

${\displaystyle E=\left(x-1\right)^{2}-4}$

${\displaystyle E=\left({\sqrt {2}}-1\right)^{2}-4}$

${\displaystyle E=\left(({\sqrt {2}})^{2}-2\times {\sqrt {2}}\times 1+1^{2}\right)-4}$

${\displaystyle E=\left(2-2{\sqrt {2}}+1\right)-4}$

${\displaystyle E=-1-2{\sqrt {2}}}$

${\displaystyle E=(x-1)^{2}-4}$

${\displaystyle E=(x-1)^{2}-2^{2}}$

${\displaystyle E=(x-1-2)(x-1+2)}$

${\displaystyle E=(x-3)(x+1)}$

${\displaystyle {\begin{cases}x+1=0\\x-3=0\end{cases}}}$

${\displaystyle {\begin{cases}x=-1\\x=3\end{cases}}}$

${\displaystyle E=(2x-3)(5-2x)-(2x-3)^{2}}$

= ${\displaystyle (10x-4x^{2}-15+6x)-(4x^{2}-12x+9)}$

= ${\displaystyle -8x^{2}+28x-24}$

${\displaystyle E=(2x-3)(5-2x)-(2x-3)2}$

= ${\displaystyle \left[2x-3\right]\left[\left(5-2x\right)-\left(2x-3\right)\right]}$

= ${\displaystyle \left[2x-3\right]\left[8-4x\right]}$

${\displaystyle {\begin{cases}2x-3=0\\-4x+8=0\end{cases}}}$

${\displaystyle {\begin{cases}2x=3\\-4x=-8\end{cases}}}$

${\displaystyle {\begin{cases}x={\frac {3}{2}}\\x=2\end{cases}}}$

${\displaystyle F=(2x+3)^{2}-(x+5)(2x+3)}$

= ${\displaystyle (4x^{2}+12x+9)-(2x^{2}+3x+10x+15)}$

= ${\displaystyle 2x^{2}-x-6}$

${\displaystyle F=(2x+3)^{2}-(x+5)(2x+3)}$

= ${\displaystyle \left[2x+3\right]\left[(2x+3)-(x+5)\right]}$

= ${\displaystyle \left[2x+3\right]\left[x-2]\right]}$

${\displaystyle {\begin{cases}2x+3=0\\x-2=0\end{cases}}}$

${\displaystyle {\begin{cases}2x=-3\\x=2\end{cases}}}$

${\displaystyle {\begin{cases}x=-{\frac {3}{2}}\\x=2\end{cases}}}$

${\displaystyle {\begin{aligned}&E=(5x-2)(x+7)+(5x-2)^{2}\\&E=(5x^{2}+35x-2x-14)+(25x^{2}-20x+4)\\&E=30x^{2}+13x-10\\\end{aligned}}}$

${\displaystyle {\begin{aligned}&E=(5x-2)(x+7)+(5x-2)^{2}\\&E=(5x-2)((x+7)+(5x-2))\\&E=(5x-2)(6x+5)\\\end{aligned}}}$

${\displaystyle {\begin{aligned}&E=(5x-2)(6x+5)\\&E=(5\times {\frac {2}{5}}-2)(6\times {\frac {2}{5}}+5)\\&E=0\times ({\frac {12}{5}}+5)\\&E=0\\\end{aligned}}}$

${\displaystyle {\begin{cases}5x-2=0\\6x+5=0\end{cases}}}$

${\displaystyle {\begin{cases}5x=2\\6x=-5\end{cases}}}$

${\displaystyle {\begin{cases}x={\frac {2}{5}}\\x={\frac {-5}{6}}\end{cases}}}$

${\displaystyle S=\{{\frac {2}{5}};{\frac {-5}{6}}\}}$

${\displaystyle {\begin{aligned}&A=(2x+3)^{2}+5(x-7)^{2}\\&A=(4x^{2}+12x+9)+5(x^{2}-14x+49)\\&A=4x^{2}+12x+9+5x^{2}-70x+245)\\&A=9x^{2}-58x+254\\\end{aligned}}}$

${\displaystyle {\begin{aligned}&B=(3x-5)^{2}-(2x-3)(2x+3)\\&B=(9x^{2}-30x+25)-(4x^{2}-9)\\&B=5x^{2}-30x+34\\\end{aligned}}}$

${\displaystyle {\begin{aligned}&A=(2x+7)^{2}+5(x-3)^{2}\\&A=(4x^{2}+28x+49)+5(x^{2}-6x+9)\\&A=4x^{2}+28x+49+5x^{2}-30x+45\\&A=9x^{2}-2x+94\\\end{aligned}}}$

${\displaystyle {\begin{aligned}&B=(5x-3)^{2}-(3x-2)(3x+2)\\&B=(25x^{2}-30x+9)-(9x^{2}-4)\\&B=16x^{2}-30x+13\\\end{aligned}}}$

P= ${\displaystyle (x-2)(2x+1)-(2x+1)^{2}}$

P= ${\displaystyle 2x^{2}+x-4x-2-(4x^{2}+4x+1)}$

P= ${\displaystyle 2x^{2}+x-4x-2-4x^{2}-4x-1)}$

P= ${\displaystyle -2x^{2}-7x-3}$

P= ${\displaystyle (x-2)(2x+1)-(2x+1)^{2}}$

P= ${\displaystyle (2x+1)\left[(x-2)-(2x+1)\right]}$

P= ${\displaystyle (2x+1)\left[-x-3\right]}$

P= ${\displaystyle (2x+1)(-x-3)}$

${\displaystyle {\begin{cases}2x+1=0\\x+3=0\end{cases}}}$

${\displaystyle {\begin{cases}x=-{\frac {1}{2}}\\x=-3\end{cases}}}$

P= ${\displaystyle (x-2)(2x+1)-(2x+1)^{2}}$

P= ${\displaystyle ({\frac {-3}{7}}-2)(2{\frac {-3}{7}}+1)-(2{\frac {-3}{7}}+1)^{2}}$

P= ${\displaystyle (-{\frac {3}{7}}-{\frac {14}{7}})(-{\frac {6}{7}}+{\frac {7}{7}})-({\frac {-6}{7}}+{\frac {7}{7}})^{2}}$

P= ${\displaystyle (-{\frac {17}{7}})({\frac {1}{7}})-({\frac {1}{7}})^{2}}$

P= ${\displaystyle -{\frac {17\times 1}{7\times 7}}-{\frac {1^{2}}{7^{2}}}}$

P= ${\displaystyle -{\frac {-17}{49}}+{\frac {-1}{49}}}$

P= ${\displaystyle {\frac {-18}{49}}}$

F = ${\displaystyle (2x-5)^{2}-x(2x-5)}$

F = ${\displaystyle 4x^{2}-20x+25-2x^{2}+5x}$

F = ${\displaystyle 2x^{2}-15x+25}$

F = ${\displaystyle (2x-5)^{2}-x(2x-5)}$

F = ${\displaystyle (2x-5)\left[(2x-5)-x\right]}$

F = ${\displaystyle (2x-5)\left[x-5\right]}$

F = ${\displaystyle (2x-5)(x-5)}$

${\displaystyle {\begin{cases}2x-5=0\\x-5=0\end{cases}}}$

${\displaystyle {\begin{cases}x={\frac {5}{2}}\\x=5\end{cases}}}$