Expressions and Equations

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These are the solutions to the NSSCO past questions on Expressions and Equations.
The TI-84 Plus CE shall be used for applicable questions.
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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} $

(1.) (a.) Find the value of 3x + 7y when x = 5 and y = 7

(b.) Factorise $50x^2 - 8y^2$ completely.

(c.) Simplify $\dfrac{2x^3 + 2x^2y}{x^2 - y^2}$

$ (a.) \\[3ex] x = 5 \\[3ex] y = 7 \\[3ex] 3x + 7y \\[3ex] = 3(5) + 7(7) \\[3ex] = 15 + 49 \\[3ex] = 64 \\[5ex] (b.) \\[3ex] 50x^2 - 8y^2 \\[3ex] = 2(25x^2 - 4y^2) ...\text{Factored by GCF} \\[3ex] = 2(5^2x^2 - 2^2y^2) \\[3ex] = 2[(5x)^2 - (2y)^2] \\[3ex] = 2(5x + 2y)(5x - 2y) ...\text{Factored by Difference of Two Squares} \\[5ex] (c.) \\[3ex] \dfrac{2x^3 + 2x^2y}{x^2 - y^2} \\[5ex] = \dfrac{2x^2(x + y)}{(x + y)(x - y)} ... \dfrac{...\text{Factored by GCF}}{...\text{Factored by Difference of Two Squares}} \\[5ex] = \dfrac{2x^2}{x - y} $

(4.) Solve for x

$ \dfrac{3x}{5} = \dfrac{9}{10} \\[5ex] $

$ \dfrac{3x}{5} = \dfrac{9}{10} \\[5ex] 3x(10) = 5(9) \\[3ex] x = \dfrac{5 * 9}{3 * 10} \\[5ex] x = \dfrac{3}{2} \\[5ex] $ Check

$x = \dfrac{3}{2}$
LHS	RHS
$ \dfrac{3x}{5} \\[5ex] 3x \div 5 \\[3ex] 3\left(\dfrac{3}{2}\right) \div 5 \\[5ex] \dfrac{9}{2} * \dfrac{1}{5} \\[5ex] \dfrac{9}{10} $	$\dfrac{9}{10}$

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