گام به گام کار در کلاس صفحه 35 درس 1 ریاضی و آمار (1) (معادلۀ درجۀ دوم)

\(\begin{array}{l}1 + \frac{8}{{{x^2}}} = \frac{4}{x} \Rightarrow 1 + \frac{8}{{{x^2}}} - \frac{4}{x} = 0 \Rightarrow \frac{{{x^2}}}{{{x^2}}} + \frac{8}{{{x^2}}} - \frac{{4x}}{{{x^2}}} = 0\\\\ \Rightarrow \frac{{{x^2} - 4x + 8}}{{{x^2}}} = 0 \Rightarrow {x^2} - 4x + 8 = 0\\\\ \Rightarrow \Delta = {b^2} - 4ac = {( - 4)^2} - 4(1)(8) = 16 - 32 = - 16 < 0\end{array}\)

معادله ریشه ندارد.

\(\begin{array}{l}\frac{{x - 2}}{{x - 4}} = \frac{{x + 1}}{{x + 3}} \Rightarrow \frac{{x - 2}}{{x - 4}} - \frac{{x + 1}}{{x + 3}} = 0\\\\ \Rightarrow \frac{{(x - 2)(x + 3)}}{{(x - 4)(x + 3)}} - \frac{{(x + 1)(x - 4)}}{{(x + 3)(x - 4)}} = 0\\\\ \Rightarrow \frac{{{x^2} + x - 6 - {x^2} + 3x + 4}}{{(x - 3)(x - 4)}} = 0 \Rightarrow \frac{{4x - 2}}{{(x - 3)(x - 4)}} = 0\\\\ \Rightarrow 4x - 2 = 0 \Rightarrow x = \frac{1}{2}\end{array}\)

\(\begin{array}{l}\frac{{24}}{{10 + m}} + 1 = \frac{{24}}{{10 - m}} \Rightarrow \frac{{24}}{{10 + m}} + 1 - \frac{{24}}{{10 - m}} = 0 \Rightarrow \\\\\frac{{24(10 - m)}}{{(10 + m)(10 - m)}} + \frac{{(10 + m)(10 - m)}}{{(10 + m)(10 - m)}} - \frac{{24(10 - m)}}{{(10 + m)(10 - m)}} = 0\\\\\frac{{240 - 24m + 100 - {m^2} - 240 - 24m}}{{(10 + m)(10 - m)}} = 0\\\\ \Rightarrow - {m^2} - 48m + 100 = 0 \Rightarrow {m^2} + 48m - 100 = 0\\\\ \Rightarrow (m + 50)(m - 2) = 0 \Rightarrow \left\{ \begin{array}{l}m = - 50\\\\m = 2\end{array} \right.\end{array}\)

\(\begin{array}{l}\frac{{y + 2}}{{y + 3}} - \frac{{{y^2}}}{{{y^2} - 9}} = 1 - \frac{{y - 1}}{{3 - y}}\\\\ \Rightarrow \frac{{y + 2}}{{y + 3}} - \frac{{{y^2}}}{{(y - 3)(y + 3)}} - 1 + \frac{{y - 1}}{{ - (3 - y)}} = 0\\\\ \Rightarrow \frac{{(y + 2)(y - 3)}}{{(y + 3)(y - 3)}} - \frac{{{y^2}}}{{(y - 3)(y + 3)}} - \frac{{(y - 3)(y + 3)}}{{(y - 3)(y + 3)}} - \\\\\frac{{(y - 1)(y + 3)}}{{(y - 3)(y + 3)}} = 0\\\\ \Rightarrow \frac{{{y^2} - y - 6 - {y^2} - {y^2} + 9 - {y^2} - 2y + 3}}{{(y - 3)(y + 3)}} = 0\\\\ \Rightarrow - 2{y^2} - 3y + 6 = 0 \Rightarrow 2{y^2} + 3y - 6 = 0\\\\ \Rightarrow \Delta = {3^2} - 4(2)( - 6) = 9 + 48 = 57 > 0\\\\ \Rightarrow y = \frac{{ - 3 \pm \sqrt {57} }}{4} \Rightarrow \left\{ \begin{array}{l}y = \frac{{ - 3 - \sqrt {57} }}{4}\\\\y = \frac{{ - 3 + \sqrt {57} }}{4}\end{array} \right.\end{array}\)

\(\begin{array}{l}\left\{ \begin{array}{l}\frac{x}{{a - x}} + \frac{{a - x}}{x} = \frac{a}{x}\\\\x = 2\end{array} \right. \Rightarrow \frac{2}{{a - 2}} + \frac{{a - 2}}{2} = \frac{a}{2}\\\\ \Rightarrow \frac{2}{{a - 2}} + \frac{{a - 2}}{2} - \frac{a}{2} = 0 \Rightarrow \frac{{2(2)}}{{2(a - 2)}} + \frac{{(a - 2)(a - 2)}}{{2(a - 2)}} - \\\\\frac{{a(a - 2)}}{{2(a - 2)}} = 0 \Rightarrow \frac{{4 + {a^2} + 4 - 4a - {a^2} + 2a}}{{2(a - 2)}} = 0\\\\ \Rightarrow - 2a + 8 = 0 \Rightarrow a = 4\end{array}\)