Singapore Math GCE O Level Additional Math: Complete Course

Explore quadratic functions defined by the highest power of x being two. Write them as ax^2 + bx + c with a nonzero and b, c possibly zero.

Explore the shape of a quadratic graph, a parabola with a vertex, showing a minimum when a is positive and a maximum when a is negative.

Learn to complete the square for a quadratic by converting ax^2+bx+c into a(x+g)^2+h, with g=b/(2a) and h=c - b^2/(4a).

Learn to solve quadratic equations using factorization or the formula method, with steps to set one side to zero, identify a, b, and c, and find real roots.

Understand the discriminant, b^2 - 4ac, and how its sign reveals two distinct real roots, one repeated root, or no real roots.

Sketch a quadratic graph by completing the square to locate the vertex, find intercepts, and use the discriminant to determine real roots and tangency.

Explore inequalities on the number line between a and b. Use less than, less than or equal to, and greater than signs with open or closed circles.

Use substitution to solve simultaneous linear equations by making one variable the subject, substituting into the other, and finding the x and y values at their intersection.

Solve simultaneous equations with one linear and one non-linear equation using substitution. Make a variable the subject, substitute into the other, and solve the resulting quadratic by factoring.

Solve simultaneous equations to find the points of intersection between graphs. The number of solution sets corresponds to the number of intersection points, as demonstrated by line and curve examples.

Explore the discriminant of ax^2+bx+c: if b^2-4ac>0 there are two real roots and two x-intercepts; if =0 there is one tangent root; if <0 there are no real roots.

Explains the discriminant concept for line–curve intersections, showing how to form a quadratic by equating y and use the discriminant to determine zero, one, or two intersection points.

Explore surds and rational exponents by applying root and power rules, with a focus on the square root as the power half and practical examples like a^(m/n).

Apply the surd simplification rule sqrt(a b) = sqrt a times sqrt b to factor a surd into simpler roots, using perfect squares like 9 or 4 to obtain integers.

Learn to add and subtract surds by combining like terms with the same base. Using x=√3 and y=√2, merge 3√3+3√3 to 6√3 and -5√2-9√2 to -14√2, with no further simplification.

Multiply surds by sqrt(a) times sqrt(b) = sqrt(ab) and sqrt(a) times sqrt(a) = a. See examples: sqrt3*sqrt5 = sqrt15, sqrt3*sqrt3 = 3, and sqrt2*sqrt3 = sqrt6; only like bases simplify.

Master rationalize denominators with surds: apply conjugates for single-term and two-term quotients, using difference-of-squares to obtain clean, simplified results.

Polynomials are expressions in x with two or more terms of different powers; degree is the highest power, and we write polynomials as p(x) for evaluation like p(5) or p(-1).

Master adding and subtracting polynomials by combining like terms. Recognize that unlike terms cannot be combined, and practice adding and subtracting terms like x^3, x^2, x, and constants.

Explore the multiplication of polynomials through expansion, distribution, and combining like terms, with examples p x and q x.

Explore identities in algebra, distinguishing true-for-all identities from equations, and learn two methods—expansion with coefficient comparison and value substitution—to find coefficients a, b, c.

Learn division with long division, using 59 divided by eight, identifying the dividend, divisor, quotient, and remainder, and showing how 59 equals seven times eight plus three for polynomial division.

This lecture covers division of polynomials, using long division with descending powers and zeros for missing terms, and coefficient comparison, to obtain the quotient and remainder.

Master the factor and remainder theorems for polynomials, learn to find remainders with divisors ax+b by evaluating p(-b/a), and verify factors via zero remainder with x-1 and 2x+1.

Examine cubic equations via the standard form y = ax^3 + bx^2 + cx + d and the factorized form y = a(x−f)(x−g)(x−h) with x-intercepts f, g, h.

Sketch cubic curves with y = a(x - f)(x - g)(x - h) from x-intercepts f, g, h and y-intercept to find a, noting a's sign shapes the max/min order.

Solve cubic equations by factoring, using roots to form factors like x minus alpha, apply the factor theorem, and decompose into a linear times a quadratic by comparing coefficients.

Solve cubic inequalities using a graphical method: factor completely, identify x-intercepts, sketch the cubic, and determine solution intervals where the expression is positive or zero.

Explore partial fractions by reversing the operation of combining fractions. Learn to split a single fraction, using a common denominator, into its partial fractions.

Explore proper and improper fractions, define polynomial degree, and distinguish when a polynomial fraction is proper or improper with clear examples.

Decompose a polynomial fraction into partial fractions by factorising the denominator into non repeated linear, repeated linear, and non repeated quadratic factors, then solve for the constants.

Learn factorial, denoted by the exclamation mark, where n! equals n×(n−1)×...×1, with examples like 3! and 7!, and using the x! function on scientific calculators.

Learn to compute n choose r using the factorial formula n!/(r!(n - r)!). 5 choose 3 equals 10, and note that n and r must be positive integers with r <= n.

Master binomial expansion using the binomial formula, n choose k coefficients, and ascending powers of x to simplify expressions like one plus x to the power of n.

Explore binomial expansion with the first four terms of (3+x)^10 and the first three terms of (2-3x)^5 in ascending powers of x, and determine the coefficient of x^2 in (1-x^2)(2-3x)^5.

Learn how to identify bases and exponents, use zero and negative powers, apply fractional exponents and roots, and combine indices when bases or exponents match for multiplication and division.

Explore logarithms by converting between exponential and logarithmic forms, using base a, exponent x, and examples like 3^2=9 and log_3 9=2 to evaluate x.

Explore common log (base ten), its notation and alias lg, and learn to use a calculator to evaluate log expressions such as log 8, log 17, and combined calculations.

Explore natural log, or ln, as log base e, and learn that e equals 2.71828; use calculators to compute ln and e^x, and apply ln e^x = x.

Master the rules of logarithms, including the addition and subtraction rules with the same base, the power and coefficient rules, and the change-of-base formula for calculator evaluation.

Learn to solve equations with unknown exponents by separating terms, applying logarithms, and bringing the exponent down to solve for x, for single-term and multi-term cases.

Learn to convert angles between degrees and radians, with 180 degrees equal to pi radians and one degree equal to pi over 180 radians.

Explore sine graphs, including the sine wave with amplitude 1 and a period of 360 degrees. Learn how horizontal and vertical scaling, translations, and reflections transform y = sin x.

Follow transformation order: translate in x, scale in x, scale in y, translate in y to obtain y = a sin(bx + c) + d, noting sine's 360-degree period.

Transforming a sine graph to y = 2 sin(3x + 60°) − 1, this lesson shows translating in x, scaling in x, scaling in y, and translating in y.

Explore cosine graphs, including y = cos x, its amplitude of 1 and period of 360 degrees, and how transformations y = a cos(bx) change amplitude and period.

Explore sketching tangent graphs, noting a 180-degree period, vertical asymptotes at x = 90(2n+1) degrees, and no amplitude or extrema, with behavior mirroring sine transformations.

Explore the four quadrants, zero degrees on the positive x axis, and apply the a s t c rule to determine sine, cosine, and tangent signs.

Learn how any angle sharing a basic acute angle alpha maps to four quadrants using symmetry, and determine the signs of sine, cosine, and tangent in each quadrant.

Explore the special values of sine, cosine, and tangent at 0, 90, 180, 270, and 360 degrees, including undefined tangents.

Learn to memorize sine, cosine, and tangent values for special angles 30, 45, and 60 degrees using two right triangles and Pythagoras.

Explore the three trigonometric functions secant, cosecant, and cotangent by defining them as the reciprocals of cosine, sine, and tangent.

Master trig identities, including tan x equals sin x over cos x and sin^2 x + cos^2 x = 1, and apply 1+ tan^2 x = sec^2 x to problems.

Learn to solve trig equations by using inverse functions to find the basic acute angle in sine, cosine, or tangent forms, identify quadrants, and compute all degree and radian solutions.

Prove trig identities by transforming the left-hand side to the right-hand side using sin^2 theta + cos^2 theta = 1 and tan theta = sin theta / cos theta.

Master advanced trigonometric equations by using identities to transform tangents into sine and cosine, then factoring and solving for sine or cosine values across quadrants, including special angles.

Explore trigonometric identities, including sine, cosine, and tangent sum and difference formulas, with plus minus patterns and practice applying these identities to questions.

Apply trigonometric identities to solve for x, a, and b using sine addition formulas, cotangent relations, and quadrant reasoning with 30° special angles.

Apply the tangent addition formula with 45 degrees to show product equals -1, then use sine and cosine sum/difference identities to derive -cot(a) and sin(3pi - x) = -cos x.

Explore double angle identities for sine, cosine, and tangent, including sin 2a = 2 sin a cos a and cos 2a = cos^2 a − sin^2 a.

Solve trigonometric identities assignments by converting to common angle forms with double-angle and tangent rules, factorizing and quadrant analysis to locate all x in 0–360 for sine, cosine, and tangent.

Use double-angle and factorization on cos 2x + sin x = 0 for x in 0 to 2π. Tan 2x = 5 with tan x to locate x in quadrants.

Explore proving identities with double-angle formulas, including secant squared x over two tangent x equals cosecant 2x and sin x over one plus cosine x equals tan(x/2).

Apply the R formula to rewrite a sine theta plus or minus b cosine theta. Set r = sqrt(a^2 + b^2) and alpha = arctan(b/a), ignoring the sign.

Use the R formula to solve equations of the form a sin x ± b cos x, choosing radian or degree mode and finding valid x in the specified interval.

Compute AB as sqrt(32) units, find the midpoint M(1,7), and determine the gradient -1 for AB.

Explore horizontal, vertical, and oblique lines in a straight line lesson, learn their equations y=k, x=k, and y=mx+c, and see examples like the x axis, y axis, and specific values.

Explore oblique lines and master three methods to find a line’s equation using gradient and y-intercept, a given point, or two points, all converging to y = mx + c.

Understand gradient from y = mx + c; it measures steepness and whether the function increases or decreases, with parallel lines sharing the gradient and perpendicular lines multiplying to -1.

Compute the gradient of AB, find its midpoint M, then use M and the perpendicular slope to derive the equation of the perpendicular bisector, ensuring equidistance from A and B.

Find the area of any figure in coordinate geometry by selecting a starting point, traversing anti-clockwise, multiplying paired coordinates along two connecting lines, subtracting the sums, and halving the result.

Solve simultaneous equations to locate the intersection points of graphs, as shown with y = x^2 and y = 2x + 3, yielding (3,9) and (-1,1).