Quadratic Equations and Linear Inequalities, Objectives, Roots of a Quadratic Equation, Solving Quadratic Equation by Factorization, Alternative Method
Doorsteptutor material for IAS is prepared by world's top subject experts: Get complete video lectures from top expert with unlimited validity: cover entire syllabus, expected topics, in full detail- anytime and anywhere & ask your doubts to top experts.
Recall that an algebraic equation of the second degree is written in general form as . It is called a quadratic equation in .
The coefficient is the first or leading coefficient, ’ is the second or middle coefficient and is the constant term (or third coefficient).
For example,
are all quadratic equations.
Objectives
After studying this lesson, you will be able to:
Solve a quadratic equation with real coefficients by factorization and by using quadratic formula;
Find relationship between roots and coefficients;
Form a quadratic equation when roots are given;
Differentiate between a linear equation and a linear inequality;
State that a planl region represents the solution of a linear inequality;
Represent graphically a linear inequality in two variables;
Show the solution of an inequality by shading the appropriate region;
Solve graphically a system of two or three linear inequalities in two variables;
Roots of a Quadratic Equation
The value which when substituted for the variable in an equation, satisfies it, is called a root (or solution) of the equation.
If be one of the roots of the quadratic equation
Then
In other words, is a factor of the quadratic equation (1)
Example:
Consider a quadratic equation
Solution:
If we substitute in ,
We get, L.H.S
L.H.S.R.H.S.
Again put in ,
We get, L.H.S
L.H.S.R.H.S.
Again put in ,
We get, L.H.SR.H.S.
and are the only values of x which satisfy the quadratic equation
There are no other values which satisfy
and are the only two roots of the quadratic equation .
Solving Quadratic Equation by Factorization
Recall that you have learnt how to factorize quadratic polynomial of the form , by splitting the middle term and taking the common factors.
Same method can be applied while solving a quadratic equation by factorization.
If and are two factors of the quadratic equation
then
Either or
The roots of the quadratic equation are .
Example:
Using factorization method, solve the quadratic equation:
Solution:
The given quadratic equation is
Splitting the middle term, we have
Or
Or
Either or
Two roots of the given quadratic equation are .
Alternative Method
The given quadratic equation is
This can be rewritten as
Or
Or
The quadratic equation has equal roots .