Free Online Preparation for CAT with Minglebox e-CAT Prep. Cover basic concepts of Equations-Simple, Special and Quadratic under Quantitative Aptitude for MBA Entrance Exam Preparation with Study material, solved examples and tests prepared by CAT coaching experts.
Linear equation in one variable A linear equation in one variable is respected as ax + b = 0, where a,b are real constants and . a is known as the coefficient of 'x' variable 'x' solution of linear equation
in one variable is x = -
any linear equation in one variable always has a solution.
Linear equation in 2 variable
Linear equation in 2 variable – Ab linear equation in 2 variable is represented as ax + by = c where a,b,c are real constants, where either a or b = 0 but not both simultaneously
equation ax + by = c has infinite number of solution for real x and y but for suitable restrictions equation may have finite number of solutions.
Simultaneous linear equations
Simultaneously linear equations in two variable
A simultaneously linear equation in 2 variable is represented as
and (2) where are real constants the above pair of equation s may
have none, one or infinite number of solutions depending on the relationship between the constants.
If then we have infinite number of solutions.
If then there is no solution
If then there is one unique solution
Equation in one variable but of higher degree. The degree of equation is the highest power of variable that exists in the equation for example
is an equation of degree 3 in one variable .
Is an equation of degree 2 in one variable .
An equation of second degree in one variable is known as quadratic equation.
A equation of the form where a,b,c are real numbers and , is known as a
quadratic equation .
Roots of a quadratic equation A value of 'x' say which makes the left hand side expression equal to right hand side that is '0' .
Therefore if there exists a number such that , then ' 'is said to be a root of the
equation . In general , any quadratic will have roots either real or imaginary .
Relationship between roots of the equation and co-efficients of the equation.
If are the roots of the equation and then (some of the roots) =
and (product of roots) =
method to solve a quadratic equation.
Completing the square method at be the given equation.
Step1 : add and subtract the quantity that is
Step 2 :- Rearrange the terms as follows
Discriminant the expression is known as the discrimination of the quadratic . Just by finding the discriminant we can decide the nature of the root
Roots real and distinct
Roots real and equal
Roots are imaginary
D>0 and a perfect square
Roots are distinct real and Rational
Factorization method Before attempting this method , check the discriminant of the equation . If the discriminant D is zero or a perfect square , then this method will easily given the roots of the equation . We illustrated this by an example.
Find the roots of
Step i :-
Find the discriminant of the equation her by compression we get a = 1, b= -5, and c = 6 the
discriminant D = which is a perfect square
Step ii :-
Consider the constant term C, and factories it into 2 factors whose product is C. here C = 6
Therefore 2 factor products of 6 are
For each pair try to see in which case the sum of factors is equal to b(the coefficient of x) here b= -5,
we see that satisfies this as -5 = (-2) + (-3)
Step iii :-
Write the coefficient of x as sum of the 2 factors obtained in step
take out the common terms for first 2 terms and do the same for last 2 terms
x (x-2) -3 (x-2)
(x-2) is common in both the terms, so pull it out we get (x-2) (x-3) = 0
Step iv :-
Equation each term equal to '0' to get the root
x-2 =0 or x-3 =0
x-2 or x-3
Hence roots of are x=2 and x=3 respectively