Advanced Math

Inequalities

Inequalities may seem scary at first, but once you understand what the question is asking, it becomes easier.

Key points:

Multiplying by a negative number flips the inequality sign.
Do NOT divide or mulitply by a variable in inequalities, as you do not know if the variable is positive or not. Instead, add or subtract the variable so the sign is unchanged.

Worked example

Non-linear functions

Many questions will have systems of equations with one quadratic equation and one linear equation. In many situations, it is very useful to use the discriminant. Remember that the discriminant is derived from the quadratic formula.

Vertex form is also useful in order to find the vertex of a parabola, where (h,k) is the vertex of the function:

Worked example

The minimum value of the given quadratic function can be found using the vertex form, where a=1. The equation can be expressed as g(x) = (x-0)^2 + 55, where (0, 55) is the global minimum point. The minimum value of the function is 55. Hence, answer choice C is correct.

A more detailed answer choice justification can be found here.

Sums and products of roots (Viete's formulae)

These formulae are useful for finding the sums and products of roots without calculating exact the values of each root

Worked example

Soultion: The given equation can be re-written into

x^2-16x+39=0

The sum of the roots is equal to -b/a=-1*(-16)/(1)=16

Questions might also be related to modelling (exponential equations)

Worked example

Modulus function (absolute value)

The modulus function's purpose is basically to measure the distance from a point to zero. In practise, this means that all negative values become positive. It is denoted as two vertical bars beside the number. For example, |-2| = 2

Worked example

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