This chapter is about quadratic expressions of the form ax2 + bx +c and their graphs. When you have completed it, you should
1. know how to complete the square in a quadratic expression
2. know how to locate the vertex and the axis of symmetry of the quadratic graph y=ax2 + bx +c (under topic functions and graphs)
3. be able to solve quadratic equations
4. know that the discriminant of the quadratic expression ax2 + bx +c is the value of b2-4ac, and know how to use it
5. be able to solve a pair of simultaneous equations involving a quadratic equation and a linear equation
6. be able to recognize and solve equations which can reduced to quadratic equations by a substitution.
Okay, i assumed you know that the equation y=bx + c has a graph which is a straight line. It is called linear equation.
NOTE : If you aren't sure about this, then read the page about FUNCTIONS AND GRAPHS before you go on.
y=ax2 + bx +c, the graph is parabola. The expression ax2 + bx +c, where a, b and c are constants, is called a quadratic. Thus, x2, x2-6x+8, 2x2-3x=4 and -3x2-5 are all examples of quadratics.
NOTE : I will point out those questions examiners are going to ask in the exams.
1. Completing the square
This is the general formula for completing square, 
You need to do completing square to get points for plotting graphs( eg : vertex, axis of symmetry and y-intercept). It the question doesn't ask you to show these points, simply press your calculator to get the roots of the equation. Beware of the sign of ax2. If a has a positive sign, then you should draw 'smile face' and 'sad face' for negative sign. 
but i found it's hard to understand this formula(for me). Let me show you a faster way to solve this type of questions.
Ex : Write x2 + 10x + 32 in completed square form.
Ex : Express 2x2 + 5x - 3 in completed square form.
here's another question that will be asked in the exams.
Find the domain and range of 2x2 + 5x - 3.
Since the answer you got is 2(x + 5/4x)2 - 49/8 , the range is simply just
y ≥ - 49/8.
For domain , x = -5/4 .
( no calculation needed. Each answer worths 1 mark.)
NOTE: Ignore the coefficient outside the bracket. if the equation is 2(x - 5/4x)2 - 49/8, so the domain would be x = 5/4
3. Solving quadratic equation
You will be familiar with solving quadratic equations of the form x2 - 6x + 8 = 0 by factorising x2 - 6x + 8 into the form (x - 2)(x - 4), and then using the argument :
if (x - 2)(x-4) = 0
x - 2 = 0 or x - 4 = 0 ( you can skip this part)
so x = 2 or 4
the solution of the equation is x = 2 or x = 4. The number 2 and 4 are the roots of the equation.
Trick : YOU CAN PRESS YOUR CALCULATOR TO GET THE ANSWER IN THE EXAMS first, then just write down the (x- ? )(x - ?).
However, some expression may not have factors, eg : 30x2 - 11x - 30.
If the question doesn't require you to give the exact value, simply press your calculator to get the answer. Do exactly the same thing i told you before.
If the examiners want you to get the exact value, then use the quadratic formula :
This type of question seldom come out in the exams, but you should know how to substitute in the values of coefficients(a,b and c) into the formula. You always could get the answers by pressing your calculator.
4. The discriminant b2- 4ac
When should i use this formula?
Well, it is used when you need to find whether the equation is larger , equal or smaller than 0 and to find the value of an unknown in the equation.
If b2 - 4ac > 0, the equation ax2 + bx + c = 0 will have 2 roots.
If b2 - 4ac < 0, there will be no roots.
If b2 - 4ac = 0, so there is one root only. (or you can say there is a repeated root)
Ex. The equation kx2 - 2x - 7 = 0 has two real roots. What can you deduct about the value of the constant k?
5. Simultaneous Equations
Ex. Solve the simultaneous equations y = x2 , x + y = 6
There is 2 ways to do this type of question. Choose the one you more comfortable with, the final answer will be the same.
6. Equations which reduce to quadratic equations
Sometimes you will come across equations which are not quadratic, don't worry, you can change it into quadratic equation, by making right substitution.
Ex. Solve the equation t4 - 13t2 + 36 = 0
This is called a quartic equation because it has a t4 term.
Quadratics
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