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корнями",{"__TSPROSE_proseElement":174,"schema":9410,"children":9412},{"name":9411,"type":281,"linkable":272,"__TSPROSE_schema":174},"captionSecondary",[9413],{"__TSPROSE_proseElement":174,"schema":9414,"data":9415},{"name":280,"type":281,"linkable":272,"__TSPROSE_schema":174},"Уравнения в одном мешке отличаются только общим множителем","image-zXy43daAZ",{"__TSPROSE_proseElement":174,"schema":9418,"children":9419,"id":9443},{"name":276,"type":271,"linkable":174,"__TSPROSE_schema":174},[9420,9423,9429,9432,9440],{"__TSPROSE_proseElement":174,"schema":9421,"data":9422},{"name":280,"type":281,"linkable":272,"__TSPROSE_schema":174},"Эта ситуация похожа на то, что любая дробь, например ",{"__TSPROSE_proseElement":174,"schema":9424,"data":9425,"storageKey":9427,"id":9428},{"name":420,"type":281,"linkable":174,"__TSPROSE_schema":174},{"katex":9426},"\\frac{1}{2}","$ \\frac{1}{2} $","inlinerMath-KErF8upF8",{"__TSPROSE_proseElement":174,"schema":9430,"data":9431},{"name":280,"type":281,"linkable":272,"__TSPROSE_schema":174},", образует ",{"__TSPROSE_proseElement":174,"schema":9433,"data":9434,"children":9435,"id":9439},{"name":8718,"type":281,"linkable":174,"__TSPROSE_schema":174},{"type":8720},[9436],{"__TSPROSE_proseElement":174,"schema":9437,"data":9438},{"name":280,"type":281,"linkable":272,"__TSPROSE_schema":174},"целый класс дробей, обозначающих одно и то же","emphasis-OiDTySXfG",{"__TSPROSE_proseElement":174,"schema":9441,"data":9442},{"name":280,"type":281,"linkable":272,"__TSPROSE_schema":174},", которые отличаются друг от друга только домножением числителя и знаменателя на одно и то же число:","paragraph-6efp8eXDQ",{"__TSPROSE_proseElement":174,"schema":9445,"data":9446,"storageKey":9448,"id":9449},{"name":499,"type":271,"linkable":174,"__TSPROSE_schema":174},{"katex":9447,"freeze":272},"\\frac{1}{2} = \\frac{2}{4} = \\frac{3}{6} = \\cdots","$$ \\frac{1}{2} = \\frac{2}{4} = \\frac{3}{6} = \\cdots $$","blockMath-r3ASxHsUT","quadratic-root-classes",{"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fincomplete":9452,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fcompleting-the-square":9459,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula":9463,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Ffactoring":9467,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas":9471,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fmental-solving":9475,"$ \\dfrac{5}{2} $":9479,"$ 2 $":9479,"$ \\dfrac{1}{2} $":9479,"$ x $":9479,"$$ x + \\frac{1}{x} = \\frac{5}{2} $$":9479,"$$ \\frac{x^2 + 1}{x} = \\frac{5}{2} $$":9479,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Felementary\u002Farticle\u002F$sameActionRule":9480,"$$ \\frac{x^2 + 1}{\\cancel{x}} \\cdot \\cancel{\\yellow{x}} = \\frac{5}{2} \\cdot \\yellow{x} \\\\ x^2 + 1 = \\frac{5x}{2} $$":9479,"$$ \\yellow{2}(x^2 + 1) = \\frac{5x}{\\cancel{2}} \\cdot \\cancel{\\yellow{2}} \\\\ 2x^2 + 2 = 5x $$":9479,"$ 5x $":9479,"$$ 2x^2 + 2 - \\yellow{5x} = \\cancel{5x} - \\cancel{\\yellow{5x}} \\\\ 2x^2 - 5x + 2 = 0 $$":9479,"$$ A = 2, \\enspace B = -5, \\enspace C = 2 $$":9479,"$$ D = B^2 - 4AC = (-5)^2 - 4\\cdot2\\cdot2 = 25 - 16 = 9 $$":9479,"$$ x_{1,2} = \\frac{-B \\pm \\sqrt{D}}{2A} = \\frac{-(-5) \\pm \\sqrt{9}}{2\\cdot2} = \\frac{5 \\pm 3}{4} \\\\ x_1 = \\frac{8}{4} = 2, \\enspace x_2 = \\frac{2}{4} = \\frac{1}{2} \\\\ \\boxed{x_1 = 2, \\enspace x_2 = \\frac{1}{2}} $$":9479,"$ 244 $":9479,"$ x + 2 $":9479,"$ -12 $":9479,"$ -10 $":9479,"$ 10 $":9479,"$ 12 $":9479,"$$ x^2 + (x + 2)^2 = 244 $$":9479,"$$ x^2 + (x^2 + 4x + 4) = 244 \\\\ 2x^2 + 4x + 4 = 244 $$":9479,"$$ 2x^2 + 4x - 240 = 0 $$":9479,"$$ x^2 + 2x - 120 = 0 $$":9479,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas\u002Farticle\u002F$vietasFormulas":9485,"$$ \\begin{cases} x_1 + x_2 = -2 \\\\ x_1 \\cdot x_2 = -120 \\end{cases} $$":9479,"$ -2 $":9479,"$ -120 $":9479,"$$ 4m^2 + m - 3 = 0 $$":9479,"$$ m_1 = \\frac{3}{4} >> m_2 = -1 $$":9479,"$$ 4\\left(m - \\frac{3}{4}\\right)(m + 1) = 0 $$":9479,"$ A $":9479,"$ C $":9479,"$$ m^2 + m - 12 = 0 $$":9479,"$$ \\begin{dcases} x_1 + x_2 = -1 \\\\ x_1x_2 = -12 \\end{dcases} $$":9479,"$ -1 $":9479,"$ -4 $":9479,"$ 3 $":9479,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fmental-solving\u002Farticle\u002F$methodAtoC":9488,"$ 4 $":9479,"$$ x_1 = -1 >> x_2 = \\frac{3}{4} $$":9479,"$$ (2m)^2 + \\yellow{2} \\cdot \\yellow{2}m \\cdot \\yellow{\\frac{1}{4}} - 3 = 0 \\\\ \\underbrace{(2m)^2 + 2m \\cdot \\frac{1}{4} + \\yellow{\\left(\\frac{1}{4}\\right)^2}}_{\\normalsize a^2 + 2\\cdot a\\cdot b + b^2} - \\yellow{\\left(\\frac{1}{4}\\right)^2} - 3 = 0 \\\\ \\left(2m + \\frac{1}{4}\\right)^2 - \\frac{1}{16} - 3 = 0 \\\\ \\left(2m + \\frac{1}{4}\\right)^2 - \\frac{49}{16} = 0 $$":9479,"$ \\frac{49}{16} $":9479,"$$ \\left(2m + \\frac{1}{4}\\right)^2 = \\frac{49}{16} $$":9479,"$$ 2m + \\frac{1}{4} = \\pm\\frac{7}{4} $$":9479,"$$ A = 4, \\enspace B = 1, \\enspace C = -3 $$":9479,"$$ D = B^2 - 4AC = 1^2 - 4\\cdot4\\cdot(-3) = 1 + 48 = 49 $$":9479,"$$ m_{1,2} = \\frac{-B \\pm \\sqrt{D}}{2A} = \\frac{-1 \\pm 7}{8} $$":9479,"$$ m_1 = \\frac{6}{8} = \\frac{3}{4} >> m_2 = -1 $$":9479,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Ffactoring\u002Farticle\u002F$generalFactoring":9491,"$$ 4\\left(x - \\frac{3}{4}\\right)(x+1) = 0 $$":9479,"$$ 2x^2 + 3x + 3 = 0 $$":9479,"$$ x^2 + 3x + 6 = 0 $$":9479,"$$ \\begin{dcases} x_1 + x_2 = -3 \\\\ x_1x_2 = 6 \\end{dcases} $$":9479,"$$ (\\sqrt{2}x)^2 + \\yellow{2} \\cdot \\yellow{\\sqrt{2}}x \\cdot 3 \\cdot \\yellow{\\frac{1}{2\\sqrt{2}}} + 3 = 0 \\\\ \\underbrace{(\\sqrt{2}x)^2 + 2 \\cdot \\sqrt{2}x \\cdot \\frac{3}{2\\sqrt{2}} + \\yellow{\\left(\\frac{3}{2\\sqrt{2}}\\right)^2}}_{\\normalsize a^2 + 2\\cdot a \\cdot b + b^2} - \\yellow{\\left(\\frac{3}{2\\sqrt{2}}\\right)^2} + 3 = 0 \\\\ \\left(\\sqrt{2}x + \\frac{3}{2\\sqrt{2}}\\right)^2 - \\frac{9}{8} + 3 = 0 \\\\ \\left(\\sqrt{2}x + \\frac{3}{2\\sqrt{2}}\\right)^2 + \\frac{15}{8} = 0 $$":9479,"$ \\frac{15}{8} $":9479,"$$ \\left(\\sqrt{2}x + \\frac{3}{2\\sqrt{2}}\\right)^2 = -\\frac{15}{8} $$":9479,"$$ A = 2, \\enspace B = 3, \\enspace C = 3 $$":9479,"$$ D = B^2 - 4AC = 3^2 - 4\\cdot2\\cdot3 = 9 - 24 = -15 $$":9479,"$$ (v+1)(v-5) = 4 $$":9479,"$$ v_1 = 2 + \\sqrt{13}, \\enspace v_2 = 2 - \\sqrt{13} $$":9479,"$$ (v - 2 - \\sqrt{13})(v - 2 + \\sqrt{13}) = 0 $$":9479,"$$ v^2 - 5v + v - 5 = 4 $$":9479,"$$ v^2 - 5v + v - 5 - 4 = 0 $$":9479,"$$ v^2 - 4v - 9 = 0 $$":9479,"$$ \\begin{dcases} x_1 + x_2 = 4 \\\\ x_1x_2 = -9 \\end{dcases} $$":9479,"$$ \\underbrace{v^2 - 2 \\cdot v \\cdot 2 + \\yellow{2^2}}_{\\normalsize a^2 - 2\\cdot a \\cdot b + b^2} - \\yellow{2^2} - 9 = 0 \\\\ (v - 2)^2 - 4 - 9 = 0 \\\\ (v - 2)^2 - 13 = 0 $$":9479,"$ 13 $":9479,"$$ (v - 2)^2 = 13 $$":9479,"$$ v - 2 = \\pm\\sqrt{13} \\\\ v = 2 \\pm \\sqrt{13} \\\\ v_1 = 2 + \\sqrt{13}, \\enspace v_2 = 2 - \\sqrt{13} $$":9479,"$$ A = 1, \\enspace B = -4, \\enspace C = -9 $$":9479,"$$ D = B^2 - 4AC = (-4)^2 - 4\\cdot1\\cdot(-9) = 16 + 36 = 52 $$":9479,"$$ v_{1,2} = \\frac{-B \\pm \\sqrt{D}}{2A} = \\frac{-(-4) \\pm \\sqrt{52}}{2\\cdot1} = \\frac{4 \\pm 2\\sqrt{13}}{2} = 2 \\pm \\sqrt{13} $$":9479,"$$ (y + 2)(y + 6) = 21 $$":9479,"$$ y_1 = 1, \\enspace y_2 = -9 $$":9479,"$$ (y - 1)(y + 9) = 0 $$":9479,"$$ (y + 2)(y + 6) = 21 \\\\ y^2 + 6y + 2y + 12 = 21 $$":9479,"$ 21 $":9479,"$$ y^2 + 6y + 2y + 12 - 21 = 0 $$":9479,"$$ y^2 + 8y - 9 = 0 $$":9479,"$$ \\begin{dcases} x_1 + x_2 = -8 \\\\ x_1 \\cdot x_2 = -9 \\end{dcases} $$":9479,"$ -8 $":9479,"$ -9 $":9479,"$ 1 $":9479,"$$ \\underbrace{y^2 + 2 \\cdot y \\cdot 4 + \\yellow{4^2}}_{\\normalsize a^2 + 2 \\cdot a \\cdot b + b^2} - \\yellow{4^2} - 9 = 0 \\\\ (y + 4)^2 - 16 - 9 = 0 \\\\ (y + 4)^2 - 25 = 0 $$":9479,"$ 25 $":9479,"$$ (y+4)^2 = 25 $$":9479,"$ y + 4 $":9479,"$$ y + 4 = \\pm 5 \\\\ y_1 = 1, \\enspace y_2 = -9 $$":9479,"$$ A = 1, \\enspace B = 8, \\enspace C = -9 $$":9479,"$$ D = B^2 - 4AC = 8^2 - 4\\cdot1\\cdot(-9) = 64 + 36 = 100 $$":9479,"$$ y_{1,2} = \\frac{-B \\pm \\sqrt{D}}{2A} = \\frac{-8 \\pm \\sqrt{100}}{2\\cdot1} = \\frac{-8 \\pm 10}{2} = -4 \\pm 5 $$":9479,"$$ y_1 = 1 >> y_2 = -9 $$":9479,"$$ \\frac{1}{3}n^2 + \\frac{1}{2} = -n $$":9479,"$$ n_1 = \\frac{\\sqrt{3} - 3}{2} >> n_2 = \\frac{-\\sqrt{3} - 3}{2} $$":9479,"$$ 2\\left(n - \\frac{\\sqrt{3} - 3}{2}\\right)\\left(n - \\frac{-\\sqrt{3} - 3}{2}\\right) = 0 $$":9479,"$ 6 $":9479,"$$ 2n^2 + 3 = -6n $$":9479,"$ 6n $":9479,"$$ 2n^2 + 6n + 3 = 0 $$":9479,"$$ n^2 + 6n + 6 = 0 $$":9479,"$$ \\begin{cases} n_1 + n_2 = -6 \\\\ n_1n_2 = 6 \\end{cases} $$":9479,"$ -6 $":9479,"$$ (\\sqrt{2}n)^2 + 2 \\cdot \\yellow{\\sqrt{2}}n \\cdot 3 \\cdot \\yellow{\\frac{1}{\\sqrt{2}}} + 3 = 0 \\\\ \\underbrace{(\\sqrt{2}n)^2 + 2 \\cdot \\sqrt{2}n \\cdot \\frac{3}{\\sqrt{2}} + \\yellow{\\left(\\frac{3}{\\sqrt{2}}\\right)^2}}_{\\normalsize a^2 + 2ab + b^2} - \\yellow{\\left(\\frac{3}{\\sqrt{2}}\\right)^2} + 3 = 0 \\\\ \\left(\\sqrt{2}n + \\frac{3}{\\sqrt{2}}\\right)^2 - \\frac{9}{2} + 3 = 0 \\\\ \\left(\\sqrt{2}n + \\frac{3}{\\sqrt{2}}\\right)^2 - \\frac{3}{2} = 0 $$":9479,"$ \\frac{3}{2} $":9479,"$$ \\left(\\sqrt{2}n + \\frac{3}{\\sqrt{2}}\\right)^2 = \\frac{3}{2} $$":9479,"$$ \\sqrt{2}n + \\frac{3}{\\sqrt{2}} = \\pm\\sqrt{\\frac{3}{2}} \\\\ \\sqrt{2}n + \\frac{3}{\\sqrt{2}} = \\pm\\frac{\\sqrt{3}}{\\sqrt{2}} $$":9479,"$ \\sqrt{2} $":9479,"$$ 2n + 3 = \\pm\\sqrt{3} $$":9479,"$$ A = 2, \\enspace B = 6, \\enspace C = 3 $$":9479,"$$ D = B^2 - 4AC = 6^2 - 4\\cdot2\\cdot3 = 36 - 24 = 12 $$":9479,"$$ n_{1,2} = \\frac{-B \\pm \\sqrt{D}}{2A} = \\frac{-6 \\pm \\sqrt{12}}{2\\cdot2} = \\frac{-6 \\pm 2\\sqrt{3}}{4} = \\frac{-3 \\pm \\sqrt{3}}{2} $$":9479,"$$ n_1 = \\frac{-3 + \\sqrt{3}}{2} >> n_2 = \\frac{-3 - \\sqrt{3}}{2} $$":9479,"$$ 2\\left(n - \\frac{-3 + \\sqrt{3}}{2}\\right)\\left(n - \\frac{-3 - \\sqrt{3}}{2}\\right) = 0 $$":9479,"$$ 463x^2 - 102x - 361 = 0 $$":9479,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fmental-solving\u002Farticle\u002F$methodZeroSum":9494,"$ 0, 1, -1, 2, -2 $":9479,"$$ x_1 = 1 >>{big} x_2 = \\frac{-361}{463} $$":9479,"$$ 463 - 102 - 361 = 0 $$":9479,"$$ x_1 = 1 >> x_2 = \\frac{-361}{463} $$":9479,"$$ 463 \\cdot 1^2 - 102 \\cdot 1 - 361 = 0 \\\\ 463 - 102 - 361 = 0 \\\\ 463 - 463 = 0 \\\\ 0 = 0 $$":9479,"$$ \\underset{x_1}{1} \\cdot x_2 = \\frac{-361}{463} \\\\ \\boxed{x_2 = \\frac{-361}{463}} $$":9479,"$$ 67 x^2 - 105x - 172 = 0 $$":9479,"$$ x_1 = -1 >>{big} x_2 = \\frac{172}{67} $$":9479,"$ B $":9479,"$$ 67 - (-105) - 172 = 0 $$":9479,"$$ x_1 = -1 >> x_2 = \\frac{172}{67} $$":9479,"$$ 67 \\cdot (-1)^2 - 105 \\cdot (-1) - 172 = 0 \\\\ 67 + 105 - 172 = 0 \\\\ 172 - 172 = 0 \\\\ 0 = 0 $$":9479,"$$ \\underset{x_1}{-1} \\cdot x_2 = \\frac{-172}{67} \\\\ \\boxed{x_2 = \\frac{172}{67}} $$":9479,"$$ 13x^2 - 33x + 14 = 0 $$":9479,"$$ x_1 = 2 >>{big} x_2 = \\frac{7}{13} $$":9479,"$$ 13 \\cdot 2^2 - 33 \\cdot 2 + 14 = 0 \\\\ 52 - 66 + 14 = 0 \\\\ -14 + 14 = 0 \\\\ 0 = 0 $$":9479,"$$ \\underset{x_1}{2} \\cdot x_2 = \\frac{14}{13} \\\\ \\boxed{x_2 = \\frac{7}{13}} $$":9479,"$$ 8x^2 + 25x + 18 = 0 $$":9479,"$$ x_1 = -2 >>{big} x_2 = -\\frac{9}{4} $$":9479,"$$ 8 \\cdot (-2)^2 + 25 \\cdot (-2) + 18 = 0 \\\\ 32 - 50 + 18 = 0 \\\\ -18 + 18 = 0 \\\\ 0 = 0 $$":9479,"$$ \\underset{x_1}{-2} \\cdot x_2 = \\frac{18}{8} \\\\ \\boxed{x_2 = -\\frac{9}{4}} $$":9479,"$$ 0, 1, -1, 2, -2 $$":9479,"$$ 2x^2 - 5x - 7 = 2 \\cdot \\left(\\frac{3}{5}\\right)^2 - 5 \\cdot \\left(\\frac{3}{5}\\right) - 7 $$":9479,"$ x_1 = \\dfrac{3}{5} $":9479,"$ x_2 = \\dfrac{19}{10} $":9479,"$ \\frac{3}{5} $":9479,"$$ 2\\cdot\\left(\\frac{3}{5}\\right)^2 - 5\\cdot\\left(\\frac{3}{5}\\right) - 7 = 2\\cdot\\left(\\frac{3}{5}\\right)^2 - 5\\cdot\\left(\\frac{3}{5}\\right) - 7 $$":9479,"$$ \\frac{3}{5} + x_2 = \\frac{5}{2} \\\\ \\frac{6}{10} + x_2 = \\frac{25}{10} \\\\ \\boxed{x_2 = \\frac{19}{10}} $$":9479,"$ \\dfrac{3}{5} $":9479,"$ \\dfrac{19}{10} $":9479,"$$ 3x^2 + 7x - 2 = 3 \\cdot \\left(-\\frac{16}{3}\\right)^2 + 7 \\cdot \\left(-\\frac{16}{3}\\right) - 2 $$":9479,"$ x_1 = -\\dfrac{16}{3} $":9479,"$ x_2 = 3 $":9479,"$ \\dfrac{-16}{3} $":9479,"$$ 3\\cdot\\left(-\\frac{16}{3}\\right)^2 + 7\\cdot\\left(-\\frac{16}{3}\\right) - 2 = 3\\cdot\\left(-\\frac{16}{3}\\right)^2 + 7\\cdot\\left(-\\frac{16}{3}\\right) - 2 $$":9479,"$$ -\\frac{16}{3} + x_2 = -\\frac{7}{3} \\\\ x_2 = \\frac{9}{3} \\\\ \\boxed{x_2 = 3} $$":9479,"$$ 4x^2 - 3x + 9 = 4 \\cdot (3.7)^2 - 3\\cdot(3.7 - 3) $$":9479,"$ x_1 = 3.7 $":9479,"$ x_2 = -\\dfrac{59}{20} $":9479,"$$ 4x^2 - 3x + 9 = 4 \\cdot (3.7)^2 - 3\\cdot(3.7) + 9 $$":9479,"$ 3.7 $":9479,"$$ 4\\cdot(3.7)^2 - 3\\cdot(3.7) + 9 = 4\\cdot(3.7)^2 - 3\\cdot(3.7) + 9 $$":9479,"$$ 3.7 + x_2 = \\frac{3}{4} \\\\ \\frac{37}{10} + x_2 = \\frac{3}{4} \\\\ \\frac{74}{20} + x_2 = \\frac{15}{20} \\\\ \\boxed{x_2 = -\\frac{59}{20}} $$":9479,"$$ 5x^2 + 10x + 3 = 5 \\cdot 4.2 \\cdot (4.2 - 2) + 3 $$":9479,"$ x_1 = -4.2 $":9479,"$ x_2 = -6.2 $":9479,"$ 4.2 $":9479,"$$ 5x^2 + 10x + 3 = 5 \\cdot (4.2)^2 - 10 \\cdot (4.2) + 3 $$":9479,"$ 10x $":9479,"$ -4.2 $":9479,"$$ 5 \\cdot (-4.2)^2 + 10 \\cdot (-4.2) + 3 = 5 \\cdot (4.2)^2 - 10 \\cdot (4.2) + 3 \\\\ 5 \\cdot (-1)^2 \\cdot (4.2)^2 - 10 \\cdot (4.2) + 3 = 5 \\cdot (4.2)^2 - 10 \\cdot (4.2) + 3 \\\\ 5 \\cdot (4.2)^2 - 10 \\cdot (4.2) + 3 = 5 \\cdot (4.2)^2 - 10 \\cdot (4.2) + 3 $$":9479,"$$ -4.2 + x_2 = -\\frac{10}{5} \\\\ -4.2 + x_2 = -2 \\\\ \\boxed{x_2 = -6.2} $$":9479,"$ 200 $":9479,"$ 40 $":9479,"$ v $":9479,"$$ t = \\frac{200}{v} $$":9479,"$$ t-1 = \\frac{200}{v + 10} $$":9479,"$ t $":9479,"$$ \\frac{200}{v} - 1 = \\frac{200}{v+10} $$":9479,"$ v(v + 10) $":9479,"$$ 200(v + 10) - v(v + 10) = 200v $$":9479,"$$ 200v + 2000 - v^2 - 10v = 200v $$":9479,"$ 200v $":9479,"$$ 2000 - v^2 - 10v = 0 $$":9479,"$$ -v^2 - 10v + 2000 = 0 $$":9479,"$$ v^2 + 10v - 2000 = 0 $$":9479,"$$ \\begin{cases} v_1 + v_2 = -10 \\\\ v_1 \\cdot v_2 = -2000 \\end{cases} $$":9479,"$ -2000 $":9479,"$ -50 $":9479,"$ k $":9479,"$$ x^2 + 3x + (k^2 - 7k + 12) = 0 $$":9479,"$ k = 3 $":9479,"$ k = 4 $":9479,"$$ x_1x_2 = k^2 - 7k + 12 $$":9479,"$$ k^2 - 7k + 12 = 0 $$":9479,"$ 7 $":9479,"$ k^2 -7k + 12 $":9479,"$$ x^2 + (k^2 + 4k - 5)x - k = 0 $$":9479,"$ k = -5 $":9479,"$ k = 1 $":9479,"$$ x_1 + x_2 = -(k^2 + 4k - 5) $$":9479,"$$ -(k^2 + 4k - 5) = 0 $$":9479,"$$ k^2 + 4k - 5 = 0 $$":9479,"$ -5 $":9479,"$ k^2 + 4k - 5 $":9479,"$ a $":9479,"$$ x^2 - 2a(x-1) - 1 = 0 $$":9479,"$ a = 1 $":9479,"$ a = \\frac{1}{2} $":9479,"$$ x^2 - 2a(x - 1) - 1 = 0 \\\\ x^2 - 2ax + 2a - 1 = 0 \\\\ x^2 - 2ax + (2a - 1) = 0 $$":9479,"$$ \\begin{cases} x_1 + x_2 = 2a \\\\ x_1x_2 = 2a - 1 \\end{cases} $$":9479,"$$ x_1 + x_2 = x_1^2 + x_2^2 $$":9479,"$ 2x_1x_2 $":9479,"$$ x_1 + x_2 + \\yellow{2x_1x_2} = x_1^2 + \\yellow{2x_1x_2} + x_2^2 \\\\ x_1 + x_2 + 2x_1x_2 = (x_1 + x_2)^2 $$":9479,"$$ 2a + 2(2a - 1) = (2a)^2 \\\\ 2a + 4a - 2 = 4a^2 \\\\ 6a - 2 = 4a^2 $$":9479,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Felementary\u002Farticle\u002F$alwaysGlobal":9497,"$$ \\frac{6a - 2}{\\yellow{2}} = \\frac{4a^2}{\\yellow{2}} \\\\ 3a - 1 = 2a^2 $$":9479,"$$ 2a^2 - 3a + 1 = 0 $$":9479,"$$ \\underset{a_1}{1}\\cdot \\ ? = \\frac{1}{2} \\\\ a_2 = \\frac{1}{2} $$":9479,"$$ x^2 + 2x + C = 0 $$":9479,"$$ x_1 = -1 + \\sqrt{1 - C} >> x_2 = -1 - \\sqrt{1 - C} $$":9479,"$ 1 - C $":9479,"$$ x^2 + 2x + C = 0 \\\\ \\underbrace{x^2 + 2 \\cdot x \\cdot 1 + \\yellow{1}}_{\\small a^2 + 2 \\cdot a \\cdot b + b^2} - \\yellow{1} + C = 0 \\\\ (x+1)^2 - 1 + C = 0 $$":9479,"$$ (x+1)^2 = 1 - C $$":9479,"$ x + 1 $":9479,"$$ x + 1 = \\pm\\sqrt{1 - C} $$":9479,"$$ x = -1 \\pm \\sqrt{1 - C} $$":9479,"$$ A = 1, \\enspace B = 2, \\enspace C = C $$":9479,"$$ D = B^2 - 4AC = 2^2 - 4\\cdot1\\cdot C = 4 - 4C = 4(1 - C) $$":9479,"$$ x_{1,2} = \\frac{-B \\pm \\sqrt{D}}{2A} = \\frac{-2 \\pm \\sqrt{4(1 - C)}}{2\\cdot1} = \\frac{-2 \\pm 2\\sqrt{1 - C}}{2} = -1 \\pm \\sqrt{1 - C} $$":9479,"$ + $":9479,"$ 3x^2 + x + 100 = 0 $":9479,"$ 0 $":9479,"$ x^2 + 2x + 1 = 0 $":9479,"$$ a^2 + 2ab + b^2 = (a+b)^2 $$":9479,"$ b $":9479,"$$ x^2 + 2x + 1 = (x + 1)^2 $$":9479,"$$ x^2 + 2x + 1 = 0 \\\\ (x+1)^2 = 0 $$":9479,"$ Ax^2 $":9479,"$ Bx $":9479,"$$ x^4 - 7x^2 - 144 = 0 $$":9479,"$ t = x^2 $":9479,"$ x^2 $":9479,"$ x^4 $":9479,"$ t^2 $":9479,"$$ x_1 = 4 >> x_2 = -4 $$":9479,"$ (x^2)^2 $":9479,"$$ (x^2)^2 - 7x^2 - 144 = 0 $$":9479,"$$ t^2 - 7t - 144 = 0 $$":9479,"$$ t_1 = 16 >> t_2 = -9 $$":9479,"$$ x^2 = 16 >> x^2 = -9 $$":9479,"$$ x^6 + 9x^3 + 8 = 0 $$":9479,"$ t = x^3 $":9479,"$$ x_1 = -2 >> x_2 = -1 $$":9479,"$ x^6 $":9479,"$ (x^3)^2 $":9479,"$$ (x^3)^2 + 9x^3 + 8 = 0 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\\frac{1}{x}\\right) = 1 $$":9479,"$$ t = x + \\frac{1}{x} $$":9479,"$$ t^2 + 2t - 2 = 1 \\\\ t^2 + 2t - 3 = 0 $$":9479,"$$ t_1 = 1 >> t_2 = -3 $$":9479,"$$ x + \\frac{1}{x} = 1 >>{big} x + \\frac{1}{x} = -3 $$":9479,"$$ x^2 + 1 = x >>{big} x^2 + 1 = -3x $$":9479,"$$ x^2 - x + 1 = 0 >>{big} x^2 + 3x + 1 = 0 $$":9479,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula\u002Farticle\u002F$formula":9500,"$ m $":9479,"$$ x^2 + (2-m)x - m - 3 = 0 $$":9479,"$ m = 1 $":9479,"$$ \\begin{cases} x_1 + x_2 = m - 2 \\\\ x_1x_2 = -m - 3 \\end{cases} $$":9479,"$$ x_1^2 + x_2^2 = \\cdots $$":9479,"$$ \\cdots = x_1^2 + \\yellow{2x_1x_2} + x_2^2 - \\yellow{2x_1x_2} = (x_1 + x_2)^2 - 2x_1x_2 = \\cdots $$":9479,"$$ \\cdots = (m - 2)^2 - 2(-m - 3) = (m - 2)^2 + 2m + 6 = m^2 - 4m + 4 + 2m + 6 = m^2 - 2m + 10 $$":9479,"$$ x_1^2 + x_2^2 = m^2 - 2m + 10 $$":9479,"$$ m^2 - 2m + 10 = m^2 - 2m + \\yellow{1} + 10 - \\yellow{1} = \\boxed{(m - 1)^2 + 9} $$":9479,"$ +9 $":9479,"$ 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x_1)(x - x_2) = 0 $$":9479,"$ A_1 $":9479,"$ A_2 $":9479,"$$ A_2 = k \\cdot A_1 $$":9479,"$$ A_1(x - x_1)(x - x_2) = 0 \\\\ \\overbrace{\\yellow{k} \\cdot A_1}^{A_2}(x - x_1)(x - x_2) = 0 \\cdot \\yellow{k} \\\\ A_2(x - x_1)(x - x_2) = 0 $$":9479,"$ \\blacksquare $":9479,"$$ \\begin{cases} x_1 + x_2 = -\\frac{B_1}{A_1} \\\\ x_1 \\cdot x_2 = \\frac{C_1}{A_1} \\end{cases} >>{big} \\begin{cases} x_1 + x_2 = -\\frac{B_2}{A_2} \\\\ x_1 \\cdot x_2 = \\frac{C_2}{A_2} \\end{cases} $$":9479,"$ x_1 + x_2 $":9479,"$ -\\frac{B_1}{A_1} $":9479,"$ -\\frac{B_2}{A_2} $":9479,"$$ -\\frac{B_1}{A_1} = x_1 + x_2 = -\\frac{B_2}{A_2} \\\\ -\\frac{B_1}{A_1} = -\\frac{B_2}{A_2} $$":9479,"$$ \\frac{B_1}{A_1} = \\frac{B_2}{A_2} $$":9479,"$$ \\frac{B_1}{A_1} = \\frac{\\overbrace{n \\cdot B_1}^{B_2}}{\\underbrace{m \\cdot A_1}_{A_2}}, \\enspace n \\neq m $$":9479,"$ B_2 $":9479,"$ B_1 $":9479,"$ n $":9479,"$$ \\frac{B_1\\cdot\\cancel{A_1}}{\\cancel{A_1}} = \\frac{n \\cdot B_1 \\cdot \\cancel{A_1}}{m \\cdot 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