Expressions and Equations

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These are the solutions to the CSEC Additional Mathematics past questions on Expressions and Equations.
When applicable, the TI-84 Plus CE calculator (also applicable to TI-84 Plus calculator) solutions are provided for some 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} = \alpha + \beta = -\dfrac{b}{a} \\[5ex] (4.)\;\; \text{Product of roots} = \alpha\beta = \dfrac{c}{a} $

Formula Sheet: List of Formulae

(1.) The equation $kx^2 + x - 15 = 10$ has roots α and β, where k ∈ W.
(a.) Determine expressions for
(i.) α + β
(ii.) αβ

(b.) Given that $\alpha^2 + \beta^2 = \dfrac{61}{4}$, use the expressions in (a.)(i.) to determine the value of k.

$ kx^2 + x - 15 = 10 \\[4ex] Comparing\;\;to\;\;ax^2 + bx + c = 0 \\[4ex] a = k \\[3ex] b = 1 \\[3ex] c = -15 \\[3ex] (i.) \\[3ex] \alpha + \beta = -\dfrac{b}{a} = -\dfrac{1}{k} \\[5ex] (ii.) \\[3ex] \alpha\beta = \dfrac{c}{a} = -\dfrac{15}{k} \\[5ex] (\alpha + \beta)^2 = (\alpha + \beta)(\alpha + \beta) \\[4ex] = \alpha^2 + \alpha\beta + \alpha\beta + \beta^2 \\[4ex] = \alpha^2 + 2\alpha\beta + \beta^2 \\[4ex] = \alpha^2 + \beta^2 + 2\alpha\beta \\[4ex] \implies \\[3ex] \alpha^2 + \beta^2 + 2\alpha\beta = (\alpha + \beta)^2 \\[4ex] \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \\[4ex] = \left(-\dfrac{1}{k}\right)^2 - 2\left(-\dfrac{15}{k}\right) \\[6ex] = \dfrac{1}{k^2} + \dfrac{30}{k} \\[5ex] \implies \\[3ex] \dfrac{1}{k^2} + \dfrac{30}{k} = \dfrac{61}{4} \\[5ex] \dfrac{1 + 30k}{k^2} = \dfrac{61}{4} \\[5ex] 61k^2 = 4(1 + 30k) \\[4ex] 61k^2 = 4 + 120k \\[4ex] 61k^2 - 120k - 4 = 0 \\[4ex] Factors\;\;are\;\;2k \;\;and\;\; -122k \\[3ex] \implies \\[3ex] 61k^2 + 2k - 122k - 4 = 0 \\[4ex] k(61k + 2) - 2(61k + 2) = 0 \\[3ex] (61k + 2)(k - 2) = 0 \\[3ex] 61k + 2 = 0 \;\;\;OR\;\;\; k - 2 = 0 \\[3ex] 61k = -2 \;\;\;OR\;\;\; k = 2 \\[3ex] k = -\dfrac{2}{61} \;\;\;OR\;\;\; k = 2 \\[5ex] \text{Because k is an element of the set of Whole Numbers}, \;\; k = 2 $

(3.) Rationalize the denominator of the expression $\dfrac{1 + \sqrt{2}}{3 - \sqrt{2}}$

$ \dfrac{1 + \sqrt{2}}{3 - \sqrt{2}} \\[5ex] = \dfrac{1 + \sqrt{2}}{3 - \sqrt{2}} * \dfrac{3 + \sqrt{2}}{3 + \sqrt{2}} \\[5ex] = \dfrac{3 + \sqrt{2} + 3\sqrt{2} + 2}{3^2 - (\sqrt{2})^2} \\[6ex] = \dfrac{5 + 4\sqrt{2}}{9 - 2} \\[5ex] = \dfrac{5 + 4\sqrt{2}}{7} $

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