IITJEE(Revision) Quadratic Equations......Have a look.
A quadratic function is any function equivalent to one of the form
Here are some examples of quadratic functions
A quadratic equation is any equation equivalent to one of the form
Here are some examples of quadratic equations
Consider the quadratic equation
A real number x will be called a solution or a root if it satisfies the equation, meaning . It is easy to see that the roots are exactly the xintercepts of the quadratic function , that is the intersection between the graph of the quadratic function with the xaxis.
Example 1: Find the roots of the equation
Solution. This equation is equivalent to
Since 1 has two squareroots , the solutions for this equation are
Example 2: Find the roots of the equation
Solution. This example is somehow trickier than the previous one but we will see how to work it out in the general case. First note that we have
Therefore the equation is equivalent to
which is the same as
Since 3 has two squareroots , we get
which give the solutions to the equation
First recall the algebraic identities
We shall use these identities to carry out the process called Completing the Square. For example, consider the quadratic function
What can be added to yield a perfect square? Using the previous identities, we see that if we put 2e=8, that is e=4, it is enough to add to generate a perfect square. Indeed we have
It is not hard to generalize this to any quadratic function of the form . In this case, we have 2e=b which yields e=b/2. Hence
Example: Use Complete the Square Method to solve
Solution.First note that the previous ideas were developed for quadratic functions with no coefficient in front of . Therefore, let divide the equation by 2, to get
which equivalent to
In order to generate a perfect square we add to both sides of the equation
Easy algebraic calculations give
Taking the squareroots lead to
which give the solutions to the equation
Many inequalities lead to finding the sign of a quadratic expression. let us discuss this problem here. Consider the quadratic function
We know that
 1
 if (double root case), then we have
In this case, the function has the sign of the coefficient a.
a<0 a>0  2
 If (two distinct real roots case). In this case, we have
where and are the two roots with . Since is always positive when and , and always negative when , we get
 has same sign as the coefficient a when and ;
 has opposite sign as the coefficient a when .
a<0 a>0  3
 If (complex roots case), then has a constant sign same as the coefficient a.
a<0 a>0
Example: Solve the inequality
Solution. First let us find the root of the quadratic equation . The quadratic formula gives
which yields x= 1 or x=2. Therefore, the expression is negative or equal to 0 when .
An Extra..........
Consider the series and its associated sequence of partial sums . We will say that is convergent if and only if the sequence is convergent. The total sum of the series is the limit of the sequence , which we will denote by
So as you see the convergence of a series is related to the convergence of a sequence. Many do some serious mistakes in confusing the convergence of the sequence of partial sums with the convergence of the sequence of numbers .
Basic Properties.
 1.
 Consider the series and its associated sequence of partial sums . Then we have the formula
for any .
This implies in particular that if we know sequence of partial sums , one may generate the numbers since we have  2.
 If the series is convergent, then we must have
In particular, if the sequence we are trying to add does not converge to 0, then the associated series is divergent.
 3.
 The geometric series
converges if and only if q<1. Moreover we have
 4.
 (Algebraic Properties of convergent series) Let and be two convergent series. Let and be two real numbers. Then the new series
is convergent and moreover we have
Example. Show that the series
is divergent, even though
Answer. Note that for any , we have
Hence we have (for the associated partial sums)
Since , then we have
which implies that the series is divergent. Indeed, we do have
since
which implies
Example. Check that the following series is convergent and find its total sum
Answer. We have
Using the above properties, we see here that we are dealing with two geometric series which are convergent. Hence the original series is convergent and we have
which gives
Example. Check that the following series is convergent and find its total sum
Answer. First we need to clean the expression (by using algebraic manipulations)
We recognize a geometric series. Since , then the series is convergent and we have
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