The Complete Course on GCSE and iGCSE Additional Mathematics

Identify quadratic functions by the highest power of x being two and express them as ax^2 + bx + c with a ≠ 0. Coefficients include fractions, decimals, root three.

Explore the parabola shape of quadratic graphs, identifying the vertex as the minimum or maximum point. The sign of a determines whether it opens upward (minimum) or downward (maximum).

Learn to complete the square for a quadratic ax^2+bx+c by rewriting as a(x+g)^2+h, with g=b/(2a) and h=c−b^2/(4a), using coefficient comparison or the formula method.

Learn to identify maximum and minimum points of a quadratic by completing the square, rewriting ax^2+bx+c as a(x+g)^2+h, and reading off coordinates and values.

Learn two reliable methods to solve quadratic equations: factorization and the formula method. Apply the zero-product form, identify a, b, c, and use the quadratic formula to find roots.

Compute the discriminant, b squared minus four ac, for quadratic equations, and determine when it yields two real roots, one repeated root, or no real roots.

Sketch quadratic graphs by completing the square to locate maximum or minimum points, intercepts, and use the discriminant to identify real roots and when the x-axis is tangent.

Explore inequalities with less than, less than or equal to, greater than, and greater than or equal to, and represent them on a number line using black and white circles.

Learn to solve linear inequalities by isolating x, just like linear equations, and remember to reverse the inequality sign when you multiply or divide by a negative number.

Master solving quadratic inequalities by moving to zero, factoring to find roots, sketching upward parabolas, and reading off interval solutions between roots including endpoints.

Master the discriminant b^2 - 4ac to determine x-intercepts and the number of real roots: two distinct when >0, one repeated when =0, and none when <0, with graph insights.

Explore simultaneous equations with two variables, learn substitution and elimination methods, distinguish linear from nonlinear cases, and understand how multiple solutions arise and are verified.

Learn to solve simultaneous linear equations by the substitution method—make a variable the subject, substitute into the other equation, and find x and y.

Master solving simultaneous equations by substitution, using a linear and a non-linear pair; make one variable the subject, substitute, and solve the resulting quadratic via factorization, yielding two solutions.

Solve the simultaneous equations to obtain the point(s) of intersection of two graphs; each solution pair (x, y) identifies an intersection, with two sets implying two points.

Use the discriminant to determine how many times a line intersects a curve—two intersections, a tangent, or no intersection—by eliminating y to form a quadratic in x.

Discover how modulus functions return a non-negative magnitude by removing negative signs, illustrated with examples like mod three, mod minus three, and mod five minus nine.

Sketch y = |a x + b| by drawing y = a x + b, finding intercepts, and reflecting the portion across the x-axis to form a V with vertex.

Explore properties of the modulus function, showing y=|x| splits into x for x≥0 and −x for x<0, and explain removing modulus by squaring when |a|=|b|.

Square both sides to solve modulus equations when both sides are nonnegative, using the rule that if the modulus of a equals the modulus of b, then a^2 equals b^2.

this lecture demonstrates solving modulus equations by splitting the inner expression into positive and negative cases, then solving each case, as shown with |x-3|=2x+1 and |1-2x|=x.

This lecture teaches solving modulus inequalities by locating intersections at x = ±c, splitting into x and -x, and applying the standard inequality rules to get the solution intervals.

Learn division using 59 divided by 8, identifying the dividend, divisor, quotient, and remainder through long division, with 59 equals 7 times 8 plus 3.

Explore division of polynomials using long division and coefficient comparison to find quotient and remainder, illustrated with 2x^4+3x^3−x+1 divided by x-1.

Explore the factor and remainder theorems for polynomials, compute remainders when dividing by ax+b, and test factors with examples such as x-1 and 2x+1.

Explore cubic equations, power three, in form y = a x^3 + b x^2 + c x + d and factorized form y = a(x - f)(x - g)(x - h) to locate x-intercepts.

Cubic graphs are sketched from intercepts using factorized form (x - f)(x - g)(x - h); determine max or min points by a's sign and derive the equation.

Learn to solve cubic equations by factoring: apply the factor theorem to find a root, factor into linear and quadratic parts, and determine roots such as 1, -1/2, and 2.

Factor and set one side to zero, then solve cubic inequalities graphically, using x-intercepts to determine where the expression is greater than zero or less than or equal to zero.

Explore functions by modeling total cost as f(x) = 0.3x + 9500, combining fixed costs and variable costs, and use x as input to compute outputs like 9530 or 9800.

Explore functions as rules linking input x to output f(x) with a teddy bear cost example. Define domain and range, and show how to denote and use f(x) for 0.3x+9500.

Learn to evaluate functions by substitution, rewrite f(x) as x minus three for calculations, and explore g with inputs like a, a^2, or 2a minus 1.

Determine domain and range by sketching the graph; the caption shows f(x)=x-3 with x>0 yielding range above -3, and all-real x cases with a minimum value.

Learn set notation for domain and range, including curly-bracket and interval forms, real values, and inclusive versus exclusive inequalities. Use union for disjoint ranges and infinity notation.

Define a one-to-one function (A11): each y value comes from a unique x, tested by the horizontal line test. The caption cites f(x)=1/x as an example and demonstrates a counterexample.

Explore inverse functions through a teddy bear factory example; derive x from cost using f(x)=0.3x+9500, and interpret f^{-1} as the cost input yielding the number of bears.

An inverse function swaps inputs and outputs, denoted f^-1, mapping total cost back to teddy bears; a function must be A11 to have an inverse.

Restrict the domain to obtain an inverse by taking a one-to-one portion of the graph; for f(x)=x^2+1, the largest m with x ≤ m is zero.

Explore relationship between a function and its inverse, swapping inputs and outputs as domain and range, with a teddy bear cost example, and reflect graphs about y=x to obtain f⁻¹.

Explore finding inverses for f(x)=3x−2 and g(x)=x^2−2 with x>0, deriving f inverse x=(x+2)/3 and g inverse x=sqrt(x+2), and noting domain from the range.

Explore composite functions, define existence criteria via range of the inner function subset of the outer's domain, and learn to derive the rule and range from right to left.

Calculate the distance between two points using Pythagoras and the distance formula, find the midpoint, and determine the gradient of the line connecting the points.

Identify three line types: horizontal (y=k), vertical (x=k), and oblique (y=mx+c) with gradient m and intercept c; apply to examples like the x-axis (y=0) and other horizontal and vertical lines.

Learn how to form the equation of oblique lines using gradient and y-intercept, and three methods to find the line from m and c, a point, or two points.

Explore how the gradient in y = mx + c measures steepness, a line increases or decreases, with parallel lines sharing the gradient and perpendicular lines multiplying to -1.

Identify the perpendicular bisector of AB as the line perpendicular to AB through its midpoint, where any point is equidistant from A and B, then compute its gradient and equation.

Find the points of intersection by solving simultaneous equations for two graphs, as shown with y = x^2 and y = 2x + 3, giving (3,9) and (-1,1).

Identify circle centers and radii from standard form (x-a)^2+(y-b)^2=r^2 or from x^2+y^2+2gx+2fy+c=0 using completing the square, then sketch the circle.

Compare the distance from the point to the circle center with the radius to decide inside, on, or outside. Use the distance formula sqrt((a-e)^2+(b-f)^2).

Explore circles and straight lines, including tangent, chord, and non-intersecting cases, solved via simultaneous equations and discriminant, with examples of tangent and chord lines and circle equations.

Explore two circles in coordinate geometry, determine when they do not intersect, touch externally or internally, or intersect at two points, by using center distance, radii, and solving circle equations.

Master converting between degrees and radians using pi, where 180 degrees equals pi radians. Practice converting common angles and rounding results to one decimal place.

Calculate arc length using s = r theta with theta in radians. Determine sector area with 1/2 r^2 theta, where theta is measured in radians.

Compute the sector area using 1/2 r^2 theta and the perimeter using arc length r theta plus two radii for radius five cm and angle 1.5 radians.

Identify the base and exponent in indices and how many times to multiply. Apply rules for zero and negative powers, fractional powers, and same-base or same-power combinations.

explore fractional exponents for thirds and roots, recalling the rule for m/n powers and focusing on the square root (power half) with examples of two-thirds and three-halves.

Learn how to simplify surds by applying the rule sqrt(a b)=sqrt a sqrt b and factoring numbers into perfect squares, enabling clean simplifications like sqrt9=3 and sqrt54=3 sqrt6.

Group like surds by their bases and add or subtract those terms together. For example, 3√3 -5√2 +3√3 -9√2 simplifies to 6√3 -14√2, since unlike surds cannot combine.

Learn multiplication of surds by applying √a × √b = √(ab) and √a × √a = a, with examples on coefficients and signs, and observe that unlike bases remain unsimplified.

Master division of surds and rationalizing the denominator through two methods: single-term denominators and two-term denominators, using radical multiplication and the difference of squares to simplify.

Explore how logarithms transform exponential equations into logarithmic form and solve for x using log base a of y, with examples like 3^2=9.

Understand common log, the base-ten logarithm, and its lg notation, then use a calculator to compute log 8 and log of 35 times 12 divided by 24.

Explore natural logarithms (ln) as log base e, introduce the constant e (2.718...), and practice evaluating ln and e^x on calculators, with examples like ln e = 1 and e^2.

Combine same-base logarithms by adding for products and subtracting for quotients. Use the power rule and change-of-base formula, then compute with log or ln.

Solve equations with unknown powers, distinguishing type one and type two cases. Apply logarithms to isolate x and evaluate results with examples.

Solve inequalities with the unknown in the exponent by type one and two methods, using logs to bring down the power, and check positivity to decide or switch the sign.

Explore the graphs of e^x and ln x, their inverse relationship, reflections about the line y=x, and key features like positivity, domain restrictions, and x-axis asymptotes.

explain that y = k e^{n x} + a has a horizontal asymptote at y = a, while the signs of n and k shape and reflect the graph, as shown by y = 23 e^{3x} - 5.

Sketch y = k ln(a x + b) by locating the vertical asymptote at x = -b/a, the domain a x + b > 0, and the intercepts.

Use natural logarithms to linearize y = k x^n, giving ln y = ln k + n ln x; read n as the gradient and ln k as the intercept.