1--- 2geometry: margin=1.5cm 3columns: 2 4header-includes: 5- \usepackage{tabularx} 6--- 7 8\pagenumbering{gobble} 9\renewcommand{\arraystretch}{1.4} 10 11# Polynomials 12 13## Quadratics 14 15\newcolumntype{R}{>{\raggedleft\arraybackslash}X} 16\begin{tabularx}{\columnwidth}{|R|l|} 17 Quadratics & $x^2 + bx + c = (x+m)(x+n)$ \\ 18 & where $mn=c, \> m+n=b$ \\ 19 Difference of squares & $a^2 - b^2 = (a - b)(a + b)$ \\ 20 Perfect squares & $a^2 \pm 2ab + b^2 = (a \pm b^2)$ \\ 21 Completing the square & \parbox[t]{5cm}{$x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ \\ $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$} \\ 22 Quadratic formula & $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ \\ 23\end{tabularx} 24 25## Cubics 26 27**Difference of cubes:** $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$ 28**Sum of cubes:** $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$ 29**Perfect cubes:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$ 30 31## Linear and quadratic graphs 32 33### Forms of linear equations 34 35$y=mx+c$ where $m$ is gradient and $c$ is $y$-intercept 36${x \over a} + {y \over b}=1$ where $m$ is gradient and $(x_1, y_1)$ lies on the graph 37$y-y_1 = m(x-x_1)$ where $(a,0)$ and $(0,b)$ are $x$- and $y$-intercepts 38 39## Line properties 40 41Parallel lines: $m_1 = m_2$ 42Perpendicular lines: $m_1 \times m_2 = -1$ 43Distance: $\vec{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ 44 45 46## Cubic graphs 47 48$$y=a(bx-h)^3 + c$$ 49 50- $m=0$ at *stationary point of inflection* (i.e. (${h \over b}, k)$) 51- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$ 52- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$ 53- in form $y=a(x-b)^2(x-c)$, touches $x$-axis at $b$, intercept at $c$ 54 55## Quartic graphs 56 57### Forms of quadratic equations 58$y=ax^4$ 59$y=a(x-b)(x-c)(x-d)(x-e)$ 60$y=ax^4+cd^2 (c \ge 0)$ 61$y=ax^2(x-b)(x-c)$ 62$y=a(x-b)^2(x-c)^2$ 63$y=a(x-b)(x-c)^3$ 64 65## Literal equations 66 67Equations with multiple pronumerals. Solutions are expressed in terms of pronumerals (parameters) 68 69## Simultaneous equations (linear) 70 71- **Unique solution** - lines intersect at point 72- **Infinitely many solutions** - lines are equal 73- **No solution** - lines are parallel 74 75### Solving $\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases}$ for one, infinite and no solutions 76 77where all coefficients are known except for one, and $a, b$ are known 78 791. Write as matrices: $\begin{bmatrix}p & q \\ r & s \end{bmatrix} 80\begin{bmatrix} x \\ y \end{bmatrix} 81 = 82\begin{bmatrix} a \\ b \end{bmatrix}$ 832. Find determinant of first matrix: $\Delta = ps-qr$ 843. Let $\Delta = 0$ for number of solutions $\ne 1$ 85 or let $\Delta \ne 0$ for one unique solution. 864. Solve determinant equation to find variable 87- *--- for infinite/no solutions: ---* 885. Substitute variable into both original equations 896. Rearrange equations so that LHS of each is the same 907. If $\text{RHS}(1) = \text{RHS}(2)$, lines are coincident (infinite solutions) 91 If $\text{RHS}(1) \ne \text{RHS}(2)$, lines are parallel (no solutions) 92 93Or use Matrix -> `det` on CAS. 94 95### Solving $\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ 96a_2 x + b_2 y + c_2 z = d_2 \\ 97a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}$ 98 99- Use elimination 100- Generate two new equations with only two variables 101- Rearrange & solve 102- Substitute one variable into another equation to find another variable 103- etc.