Expressions and Equations

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These are the solutions to the GCSE past questions on Expressions and Equations.
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These are the notable notes regarding factoring

Factoring Formulas

$ \underline{Difference\;\;of\;\;Two\;\;Squares} \\[3ex] (1.)\;\;x^2 - y^2 = (x + y)(x - y) \\[5ex] \underline{Difference\;\;of\;\;Two\;\;Cubes} \\[3ex] (2.)\;\; x^3 - y^3 = (x - y)(x^2 + xy + y^2) \\[5ex] \underline{Sum\;\;of\;\;Two\;\;Cubes} \\[3ex] (3.)\;\; x^3 + y^3 = (x + y)(x^2 - xy + y^2) \\[4ex] $

Formulas Relating to Quadratic Expressions and Equations

$ (1.)\;\; Discriminant = b^2 - 4ac \\[5ex] (2.)\;\; \text{Quadratic Formula}:\;\; x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\[6ex] (3.)\;\; \text{Sum of roots} = -\dfrac{b}{a} \\[5ex] (4.)\;\; \text{Product of roots} = \dfrac{c}{a} $

(11.) Use the quadratic formula to solve the equation $\dfrac{1}{x - 2} + \dfrac{1}{3x - 7} = 1$.
Give your answers correct to 2 decimal places.
You must show all your working.

$ \dfrac{1}{x - 2} + \dfrac{1}{3x - 7} = 1 \\[5ex] LCD = (x - 2)(3x - 7) \\[3ex] \text{Multiply both sides by the LCD} \\[3ex] (x - 2)(3x - 7)\left(\dfrac{1}{x - 2}\right) + (x - 2)(3x - 7)\left(\dfrac{1}{3x - 7}\right) = (x - 2)(3x - 7)(1) \\[5ex] 1(3x - 7) + 1(x - 2) = (x - 2)(3x - 7) \\[3ex] 3x - 7 + x - 2 = 3x^2 - 7x - 6x + 14 \\[3ex] 4x - 9 = 3x^2 - 13x + 14 \\[3ex] 3x^2 - 13x + 14 = 4x - 9 \\[3ex] 3x^2 - 13x - 4x + 14 + 9 = 0 \\[3ex] 3x^2 - 17x + 23 = 0 \\[3ex] \text{Compare to: } ax^2 + bx + c = 0 \\[3ex] a = 3 \\[3ex] b = -17 \\[3ex] c = 23 \\[3ex] x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\[5ex] x = \dfrac{-(-17) \pm \sqrt{(-17)^2 - 4(3)(23)}}{2(3)} \\[5ex] = \dfrac{17 \pm \sqrt{289 - 276}}{6} \\[5ex] = \dfrac{17 \pm \sqrt{13}}{6} \\[5ex] = \dfrac{17 \pm 3.605551275}{6} \\[5ex] = \dfrac{17 + 3.605551275}{6} \;\;\;OR\;\;\; \dfrac{17 - 3.605551275}{6} \\[5ex] = \dfrac{20.60555128}{6} \;\;\;OR\;\;\; \dfrac{13.39444872}{6} \\[5ex] = 3.434258547 \;\;\;OR\;\;\; 2.232408121 \\[3ex] \approx 3.43 \;\;\;OR\;\;\; 2.23 ...\text{to 2 decimal places} \\[3ex] $ Check

$x = 3.434258547 \;\;\;OR\;\;\; 2.232408121$
LHS	RHS
$ \text{Testing for }x = 3.434258547 \\[3ex] \dfrac{1}{x - 2} + \dfrac{1}{3x - 7} \\[5ex] \dfrac{1}{3.434258547 - 2} + \dfrac{1}{3(3.434258547) - 7} \\[5ex] = \dfrac{1}{1.434258547} + \dfrac{1}{10.30277564 - 7} \\[5ex] = \dfrac{1}{1.434258547} + \dfrac{1}{3.30277564} \\[5ex] = 0.6972243617 + 0.3027756375 \\[3ex] = 0.9999999992 \\[3ex] \approx 1 $ $ \text{Testing for }x = 2.232408121 \\[3ex] \dfrac{1}{x - 2} + \dfrac{1}{3x - 7} \\[5ex] \dfrac{1}{2.232408121 - 2} + \dfrac{1}{3(2.232408121) - 7} \\[5ex] = \dfrac{1}{0.232408121} + \dfrac{1}{6.697224363 - 7} \\[5ex] = \dfrac{1}{0.232408121} + \dfrac{1}{-0.302775637} \\[5ex] = 4.302775633 + -3.302775646 \\[3ex] = 0.9999999872 \\[3ex] \approx 1 $	1

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