Solving quadratic equations
IF YOU have been following the materials presented in the last three lessons, you should realise by now that the following methods are commonly used to solve quadratic equations. These are:
■ Quadratic factors
■ Quadratic formula
■ Completion of squares
Learning each method is important. It is also critical that you know when to use the different methods. Let us review the materials presented previously with this in mind.
■ Only some quadratic equations can be solved by the factorisation method.
■ Given the equation, you should first attempt the factorisation method, unless otherwise directed.
■ If a specific method is requested, you must obey the instructions, or you will be penalised.
■ All quadratic equations with real roots (equations with real numbers as their solutions) can be solved using the formula method or completion of squares methods.
■ Be sure to use the correct formula, and be careful in processing the negative signs while using the formula method.
■ If you are asked to solve a quadratic equation correct to two decimal places, then you should use the formula method. Please be sure to practise all three methods.
Please continue to practise solving quadratic equations by attempting the following:
1. Solve the equation: t2 – 4t – 32 = 0
2. Solve the quadratic equation: 3x2 + 10x – 8 = 0, giving your answer correct to two decimal places.
3. Solve: 4x2 = 8x – 3 4. Solve the quadratic equation: x2 – 7x = – 6 5. Solve the equation, 2y2 – 11y + 15 = 0
6. Solve the simultaneous equations: x – 2y = 8 3x + 5y = 9
Let us now continue with the review of aspects of functions and relations.
POINTS TO NOTE: (with respect to the Cartesian diagram)
■ DOMAIN refers to x values.
■ RANGE refers to y values.
■ FUNCTION is a relation in which each element in the domain (x values) is mapped on to one, and only one, element in the range (y values).
■ FUNCTION is usually denoted by the symbols f or g. If y is a function of x, then the function of x is denoted as f(x) or g(x). If y is defined such that y = 4x + 1, then this is represented as follows:
y = f(x) = 4x + 1 or f : x ® 4x + 1
The latter means: The function f such that x is mapped on to 4x + 1.
The function is represented on the Cartesian diagram by a plot of the equation y = 4x + 1. All rules related to graphs and which were indicated previously must be observed.
IMAGE OF X
This is the value of f(x) for a given value of x. It is found by either reading the value off the graph or by substituting into the equation.
EXAMPLE
Given that f(x) = 3x + 5, calculate f(– 2). [f(– 2) is the value of f(x) for which x = – 2].
Since f(x) = 3x + 5
■ f(– 2) = 2 x – 2 + 5 = – 4 + 5 = – 1.
■ f(– 2) = – 1.
Note that – 4 is substituted for x in f(x).
Now please try the following:
The function g is defined by g: x ––– 2x2 + 1, find g(3).
If your answer is 19, then you are correct.
COMPOSITE FUNCTION
Given the functions f(x) and g(x), then the composite function f g(x) is the function obtained by the function g(x) being initially applied, followed by function f (x). In evaluating the composite function, we determine the function g(x), which is then substituted for x in f(x).
POINTS TO NOTE
■ It is important to note that: for f g(x), g(x) replaces x in f(x), while for g f(x), then f(x) replaces x in g(x).
NOTE THE ORDER WELL.
A common error made by some students is to find the product of f(x) and g(x). Avoid this, please.
This topic is fairly routine and so all students are encouraged to take full advantage of the marks allotted to this problem. In this regard, please attempt the following: