جواب تمرین صفحه 21 درس 1 ریاضی و آمار دهم انسانی (معادلۀ درجۀ دوم)

 الف

\(\begin{array}{l}2{x^2} - 8 = 0\\\\ \Rightarrow 2{x^2} = 8\\\\ \Rightarrow {x^2} = 4\\\\ \Rightarrow x = \pm 2\end{array}\)

 ب

\(\begin{array}{l}(x + 2)(x - 3) = x - 3\\\\ \Rightarrow (x + 2)(x - 3) - (x - 3) = 0\\\\ \Rightarrow (x - 3)(x + 2 - 1) = 0\\\\ \Rightarrow (x - 3)(x + 1) = 0\\\\ \Rightarrow \left\{ \begin{array}{l}x = - 1\\\\x = 3\end{array} \right.\end{array}\)

 پ

\(\begin{array}{l}{x^4} - 2{x^2} = 0\\\\ \Rightarrow {x^2}({x^2} - 2) = 0\\\\ \Rightarrow \left\{ \begin{array}{l}{x^2} = 0 \Rightarrow x = 0\\\\{x^2} = 2 \Rightarrow x = \pm \sqrt 2 \end{array} \right.\end{array}\)

 ت

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

 ث

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

 ج

\(\begin{array}{l}{x^2} - 5x + 6 = 0\\\\ \Rightarrow (x - 2)(x - 3) = 0\\\\ \Rightarrow \left\{ \begin{array}{l}x = 2\\\\x = 3\end{array} \right.\end{array}\)

 چ

\(\begin{array}{l}\frac{{{x^2}}}{3} = x\\\\ \Rightarrow {x^2} = 3x\\\\ \Rightarrow {x^2} - 3x = 0\\\\ \Rightarrow x(x - 3) = 0\\\\ \Rightarrow \left\{ \begin{array}{l}x = 0\\\\x = 3\end{array} \right.\end{array}\)

 ح

\(\begin{array}{l}{x^2} = 5 - {x^2}\\\\ \Rightarrow 2{x^2} = 5\\\\ \Rightarrow {x^2} = \frac{5}{2}\\\\ \Rightarrow x = \pm \sqrt {\frac{5}{2}} \\\\ \Rightarrow x = \pm \frac{{\sqrt {10} }}{2}\end{array}\)

 خ

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

 د

\(\begin{array}{l}9{x^2} + 3x - 2 = 0\\\\ \Rightarrow (3x + 2)(3x - 1) = 0\\\\ \Rightarrow \left\{ \begin{array}{l}x = \frac{1}{3}\\\\x = - \frac{2}{3}\end{array} \right.\end{array}\)

 ذ

\(\begin{array}{l}{(x - 3)^2} = 4\\\\ \Rightarrow x - 3 = \pm 2\\\\ \Rightarrow \left\{ \begin{array}{l}x - 3 = 2 \Rightarrow x = 5\\\\x - 3 = - 2 \Rightarrow x = 1\end{array} \right.\end{array}\)

\( = {6^2} = 36\) مساحت مربع

مساحت بریده شده برابر است با:

\( = {S_1} = {x^2}\) مساحت مربع کوچک

\( = {S_2} = x \times 2x = 2{x^2}\) مساحت مستطیل

\( = {S_3} = \frac{{\sqrt 2 x \times \sqrt 2 x}}{2} = {x^2}\) مساحت مثلث

\({S_1} + {S_2} + {S_3} = {x^2} + 2{x^2} + {x^2} = 4{x^2}\)

\( = 36 - 24 = 12 = 4{x^2}\) مساحت قسمت بریده شده

\( \Rightarrow {x^2} = 3 \Rightarrow x = \pm \sqrt 3 \)

\(x = - \sqrt 3 \) غیر قابل قبول است چون اندازه ضلع منفی نمی شود. بنابراین طول ضلع کوچک بریده شده برابر \(x = \sqrt 3 \) می باشد.

\(\begin{array}{l}\left\{ \begin{array}{l}x = 2 \Rightarrow x - 2 = 0\\\\x = - 3 \Rightarrow x + 3 = 0\end{array} \right.\\\\ \Rightarrow (x - 2)(x + 3) = 0\\\\ \Rightarrow {x^2} + x - 6 = 0\end{array}\)

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

الف 

\(\begin{array}{l}{(x - 1)^2} = K\\\\ \Rightarrow x - 1 = \pm \sqrt K \\\\ \Rightarrow x = 1 \pm \sqrt K \\\\K = 2 \Rightarrow x = 1 \pm \sqrt 2 \Rightarrow \left\{ \begin{array}{l}x = 1 + \sqrt 2 \\\\x = 1 - \sqrt 2 \end{array} \right.\\\\K = 4 \Rightarrow x = 1 \pm \sqrt 4 = 1 \pm 2 \Rightarrow \left\{ \begin{array}{l}x = 3\\\\x = - 1\end{array} \right.\\\\K = 0 \Rightarrow x = 1 \pm \sqrt 0 \Rightarrow x = 1\\\\K = - 9 \Rightarrow x = 1 \pm \sqrt { - 9} \otimes \end{array}\)

به ازای K=-9 جواب ندارد.

ب

به ازای K=0

پ

به ازای K>0 (مقادیر مثبت K)

ت

به ازای K<0 (مقادیر منفی K)

الف

مساحت مستطیل سفید + مساحت مثلث سفید = مساحت فلش

\( = \frac{1}{2} \times 3x \times 2x + x \times 2x = 5{x^2}\)

هزینه برچسب سفید:

\(5{x^2} \times 30 = 150{x^2}\)

فلش – مستطیل تابلو = مساحت قسمت آبی

\( = 60 \times 40 - 5{x^2} = 2400 - 5{x^2}\)

هزینه برچسب آبی:

\(\begin{array}{l}(2400 - 5{x^2}) \times 10 = 24000 - 50{x^2}\\\\ \Rightarrow 150{x^2} + 24000 - 50{x^2} = 27000\end{array}\)

 27000 = مجموع هزینه ها

\(100{x^2} = 3000 \Rightarrow {x^2} = 30 \Rightarrow x = \pm \sqrt {30} \)

\(x = - \sqrt {30} \) غیر قابل قبول است.

ب

\(\begin{array}{l}{a^2} = {(x + \frac{x}{2})^2} + {(2x)^2} = \frac{{25}}{4}{x^2}\\\\ \Rightarrow a = \pm \frac{5}{2}x\\\\ \Rightarrow a = \frac{5}{2}x\end{array}\)

\( = 2a + 3x + 2(2x) = 2(\frac{5}{2}x) + 3x + 4x = 12x\) محیط فلش

\( = 12x \times 100 = 1200x\) هزینه برچسب زرد

\( = 150{x^2} + 24000 - 50{x^2} + 1200x = 34800\) هزینه رنگ آمیزی تابلو

\(\begin{array}{l} \Rightarrow 10\,0{x^2} + 1200x - 10800 = 0\\\\ \Rightarrow {x^2} + 12x - 108 = 0\\\\ \Rightarrow (x + 18)(x - 6) = 0\\\\ \Rightarrow x = 6\end{array}\)

جواب x=-18 غیر قابل قبول می باشد.