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Linear equation

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

 

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.

 

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

 

D>0

Roots real and distinct

D=0

Roots real and equal

D<0

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.

 

Example

Question

Find the roots of

Solution

 

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

So

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

 
 
 

 

 

 

 

 

 

 

 

 

Refresher Test - Equations - Simple, Special and Quadratic Launch Test Discuss

Practice Test - Equations - Simple, Special and Quadratic Launch Test Discuss

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