Quadratic Equation Class 11 Mathematics Solutions | Exercise - 6.3
Chapter - 6
Quadratic Equation
Generally, polynomial of the second degree is called a quadratic polynomial. It means that the highest exponent of this function is 2. The standard form of a quadratic equation is y = ax²+ bx + c, where a, b, and c are numbers and a cannot be 0. Examples of quadratic equations include all of these: y = x²+ 4x + 2, y = 2x² + 4x - 9, y= x² etc.
In this chapter, we will learn different methods of solving a quadratic equation apart from using quadratic formula. It is better to learn some formulas and important topics related to quadratic equation before moving towards the solution section. So let's have a brief introduction of the quadratic equation chapter.
Introduction
In algebra, a quadratic equation is any equation that can be rearranged in standard form y = ax²+ bx + c, where x represents an unknown value, and a, b, and c represent known numbers, where a ≠ 0. If b = 0, the equation is known as the pure quadratic equation and if b ≠ 0, then the equation is known as the adfected quadratic
Roots of a Quadratic Equation
The two values of x which are obtained by solving the quadratic equation is known as the roots of a quadratic equation. These roots of the quadratic equation are also called the zeros of the equation. For example, the roots of the equation x² - 3x - 4 = 0 are x = -1 and x = 4 because each of them satisfies the equation. i.e.
- At x = -1, (-1)² - 3(-1) - 4 = 1 + 3 - 4 = 0
- At x = 4, (4)² - 3(4) - 4 = 16 - 12 - 4 = 0
Common Roots
In this section, we shall obtain conditions under which two given quadratic equations may have one root common or both roots common.
a) One Root Common
Let ax²+ bx + c = 0 and a'x² + b'x + c = 0 be two equations. Suppose that α is a root common to both the equations. Then
aα² + bα + c = 0
a'α² + b'α + c = 0
By the rule of cross-multiplication, we get
α²/bc' - b'c = α/ca' - c'a = 1/ab' - a'b
This gives α = (bc' - b'c)/(ca' - c'a) and also α = (ca' - c'a)/(ab' - a'b)
Combining the two, we have the required condition
(bc' - b'c) (ab' - a'b) = (ca' - c'a)²
and the common root is
(bc - b'c) / (ca' - c'a) Or (ca' - c'a) / (ab' - a'b)
b) Two Roots Common
If the quadratic equations have both roots common and if α and β be the common roots, then
α + β = -b/a = -b'/a Or a/a' = b/b'
and
αβ = c/a = c'/a' Or a/a' = c/c'
Hence, a/a' = b/b' = c/c',
which is the required condition for the equations to have both roots common.
Exercise - 6.3
In this PDF, you'll only find the solution of class 11 quadratic equation chapter. It contains all the solutions of exercise- 5.3. If you want the solutions of other exercises then you'll find them above this PDF. Just click on the button and you'll reach your destination.
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Is Class 11 Mathematics Guide Helpful For Student ?
I have published this Notes for helping students who can't solve difficult maths problems. Student should not fully depend on this note for completing all the exercises. If you totally depend on this note and simply copy as it is then it may affect your study.
Student should also use their own will power and try to solve problems themselves. You can use this mathematics guide PDF as a reference. You should check all the answers before copying because all the answers may not be correct. There may be some minor mistakes in the note, please consider those mistakes.
How to secure good marks in Mathematics ?
As, you may know I'm also a student. Being a student is not so easy. You have to study different subjects simultaneously. From my point of view most of the student are weak in mathematics. You can take me as an example, I am also weak in mathematics. I also face problems while solving mathematics questions.
If you want to secure good marks in mathematics then you should practise them everyday. You should once revise all the exercise which are already taught in class. When you are solving maths problems, start from easy questions that you know already. If you do so then you won't get bored.
Maths is not only about practising, especially in grade 11 you to have the basic concept of the problem. When you get the main concept of the problem then you can easily any problems in which similar concept are applied.
When your teacher tries to make the concept clear by giving examples then all students tries to remember the same example but you should never do that. You can create your own formula which you won't forget later.
If you give proper time for your practise with proper technique then you can definitely score a good marks in your examination.
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