📊 التمرين رقم 1

$$\left(\frac{7}{3}x - 2\right)\left(\frac{10}{3}x - 2\right)$$ $$= \frac{7}{3}x \times \frac{10}{3}x - \frac{7}{3}x \times 2 - 2 \times \frac{10}{3}x + 2 \times 2$$ $$= \frac{70}{9}x^2 - \frac{14}{3}x - \frac{20}{3}x + 4$$ $$= \frac{70}{9}x^2 - \frac{34}{3}x + 4$$

$$F = \frac{5}{9}x - \frac{1}{3} - \left(\frac{70}{9}x^2 - \frac{34}{3}x + 4\right)$$ $$F = \frac{5}{9}x - \frac{1}{3} - \frac{70}{9}x^2 + \frac{34}{3}x - 4$$ $$F = -\frac{70}{9}x^2 + \frac{5}{9}x + \frac{34}{3}x - \frac{1}{3} - 4$$ $$F = -\frac{70}{9}x^2 + \frac{5 + 102}{9}x - \frac{13}{3}$$ $$F = -\frac{70}{9}x^2 + \frac{107}{9}x - \frac{13}{3}$$

$$F = -\frac{70}{9}x^2 + \frac{107}{9}x - \frac{13}{3}$$ مع $$x = \frac{1}{5}$$: $$F = -\frac{70}{9} \times \left(\frac{1}{5}\right)^2 + \frac{107}{9} \times \frac{1}{5} - \frac{13}{3}$$ $$F = -\frac{70}{9} \times \frac{1}{25} + \frac{107}{45} - \frac{13}{3}$$ $$F = -\frac{70}{225} + \frac{107}{45} - \frac{13}{3}$$ $$F = -\frac{14}{45} + \frac{107}{45} - \frac{195}{45}$$ $$F = \frac{-14 + 107 - 195}{45} = \frac{-102}{45} = -\frac{34}{15}$$

$$\frac{5}{9}x - \frac{1}{3}$$ توحيد المقامات: $$= \frac{5x}{9} - \frac{3}{9} = \frac{5x - 3}{9}$$ إخراج العامل المشترك: $$= \frac{1}{9}(5x - 3)$$

$$\left(\frac{5}{3}x - 1\right)\left(\frac{13}{3} - \frac{14}{3}x\right)$$ $$= \frac{5}{3}x \times \frac{13}{3} - \frac{5}{3}x \times \frac{14}{3}x - 1 \times \frac{13}{3} + 1 \times \frac{14}{3}x$$ $$= \frac{65}{9}x - \frac{70}{9}x^2 - \frac{13}{3} + \frac{14}{3}x$$ $$= -\frac{70}{9}x^2 + \frac{65}{9}x + \frac{14}{3}x - \frac{13}{3}$$ $$= -\frac{70}{9}x^2 + \frac{65 + 42}{9}x - \frac{13}{3}$$ $$= -\frac{70}{9}x^2 + \frac{107}{9}x - \frac{13}{3}$$

$E = 6x\left(\frac{5}{3}x - 1\right)$ $E = 6x \times \frac{5}{3}x - 6x \times 1$ $E = \frac{30}{3}x^2 - 6x$ $E = 10x^2 - 6x$ لذا يمكن كتابة: $E = 6x\left(\frac{5}{3}x - 1\right)$

$$G = F + \frac{10}{3}x^2 - 2x$$ $$G = -\frac{70}{9}x^2 + \frac{107}{9}x - \frac{13}{3} + \frac{10}{3}x^2 - 2x$$ $$G = x^2\left(-\frac{70}{9} + \frac{30}{9}\right) + x\left(\frac{107}{9} - 2\right) - \frac{13}{3}$$ $$G = x^2\left(-\frac{40}{9}\right) + x\left(\frac{107 - 18}{9}\right) - \frac{13}{3}$$ $$G = -\frac{40}{9}x^2 + \frac{89}{9}x - \frac{13}{3}$$ لنحاول التفكيك بطريقة أخرى: $$G = F + \frac{10}{3}x^2 - 2x$$ باستخدام التفكيك المعروف لـ F: $$G = \left(\frac{5}{3}x - 1\right)\left(\frac{13}{3} - \frac{14}{3}x\right) + \frac{10}{3}x^2 - 2x$$

$$\left(\frac{5}{3}x - 1\right)\left(\frac{13}{3} - \frac{14}{3}x\right) = 0$$ إما: $$\frac{5}{3}x - 1 = 0 \Rightarrow x = \frac{3}{5}$$ أو: $$\frac{13}{3} - \frac{14}{3}x = 0 \Rightarrow \frac{14}{3}x = \frac{13}{3} \Rightarrow x = \frac{13}{14}$$

$E = F \Rightarrow E - F = 0$ نعوض قيم E و F: $E - F = 10x^2 - 6x - \left[\left(\frac{5}{3}x - 1\right)\left(\frac{13}{3} - \frac{14}{3}x\right)\right]$ نعلم أن: $E = 10x^2 - 6x = 6x\left(\frac{5}{3}x - 1\right)$ إذن: $E - F = 6x\left(\frac{5}{3}x - 1\right) - \left(\frac{5}{3}x - 1\right)\left(\frac{13}{3} - \frac{14}{3}x\right)$ نخرج العامل المشترك $\left(\frac{5}{3}x - 1\right)$: $E - F = \left(\frac{5}{3}x - 1\right)\left[6x - \left(\frac{13}{3} - \frac{14}{3}x\right)\right]$ $= \left(\frac{5}{3}x - 1\right)\left[6x - \frac{13}{3} + \frac{14}{3}x\right]$ $= \left(\frac{5}{3}x - 1\right)\left[\frac{18x + 14x}{3} - \frac{13}{3}\right]$ $= \left(\frac{5}{3}x - 1\right)\left[\frac{32x - 13}{3}\right]$ $= \frac{1}{3}\left(\frac{5}{3}x - 1\right)(32x - 13)$

من $E - F = 0$: $\frac{1}{3}\left(\frac{5}{3}x - 1\right)(32x - 13) = 0$ إما: $\frac{5}{3}x - 1 = 0 \Rightarrow x = \frac{3}{5}$ أو: $32x - 13 = 0 \Rightarrow x = \frac{13}{32}$

E و F متقايلان يعني: $E = -F \Rightarrow E + F = 0$ $E + F = 6x\left(\frac{5}{3}x - 1\right) + \left(\frac{5}{3}x - 1\right)\left(\frac{13}{3} - \frac{14}{3}x\right)$ نخرج العامل المشترك $\left(\frac{5}{3}x - 1\right)$: $E + F = \left(\frac{5}{3}x - 1\right)\left[6x + \left(\frac{13}{3} - \frac{14}{3}x\right)\right]$ $= \left(\frac{5}{3}x - 1\right)\left[6x + \frac{13}{3} - \frac{14}{3}x\right]$ $= \left(\frac{5}{3}x - 1\right)\left[\frac{18x - 14x}{3} + \frac{13}{3}\right]$ $= \left(\frac{5}{3}x - 1\right)\left[\frac{4x + 13}{3}\right]$ $= \frac{1}{3}\left(\frac{5}{3}x - 1\right)(4x + 13)$

من $E + F = 0$: $\frac{1}{3}\left(\frac{5}{3}x - 1\right)(4x + 13) = 0$ إما: $\frac{5}{3}x - 1 = 0 \Rightarrow x = \frac{3}{5}$ أو: $4x + 13 = 0 \Rightarrow x = -\frac{13}{4}$

مساحة المستطيل MNPQ: $$S = MN \times MP = \left(\frac{5}{3}x - 1\right) \times 1 = \frac{5}{3}x - 1$$ مساحة المستطيل ABCD: $$S' = AB \times AD = \left(\frac{7}{3}x - 2\right) \times \left(\frac{5}{3}x - 1\right)$$

$$\frac{5}{3}x - 1 = 2 \times \left(\frac{7}{3}x - 2\right) \times \left(\frac{5}{3}x - 1\right)$$ $$\frac{5}{3}x - 1 = 2\left(\frac{7}{3}x - 2\right)\left(\frac{5}{3}x - 1\right)$$ بقسمة على $$\left(\frac{5}{3}x - 1\right)$$ (إذا كان ≠ 0): $$1 = 2\left(\frac{7}{3}x - 2\right)$$ $$1 = \frac{14}{3}x - 4$$ $$\frac{14}{3}x = 5$$ $$x = \frac{15}{14}$$