Understanding equations: types and solutions

Equation: If two mathematical expressions are connected with the symbol of Equality then it is called equation.

Examples: 2x+5 = 3x-2

Note: The best example of equation is the weighing machine.

Degree of equation: The highest exponent of variable involved in the given equation is called the degree of equation.

Types of Equation: 

1. Linear Equation

2. Quadratic Equation 

3. Cubic Equation

4. Bi Quadratic Equation

Quadratic Equation: If the highest exponent of variables in the given equation is of 2 then it is called the Quadratic Equation.

The general form of quadratic Equation is ax² + bx+ c= 0 

Where a≠0 and a,b, c are the literal coefficients.

Conditions 

1.If b=0 then ax²+ c = 0 is called the pure Quadratic Equation.

2. If a= 0 then bx+c =0 is called the linear Equation.

Solution of Quadratic Equations:

Quadratic Equation can be solved by three ways 

i. By Making factors of middle term

ii. By completing square

iii. By using Quadratic formula

By making factors of middle term: 

Type 1

By multiplying first and the third terms and answer is positive then factors will be both positive or both negative.

Examples:i. 2x²+ 5x+3= 0 factors of 6x² = (2x)(3x)

2x²+ 2x + 3x+ 3= 0 

2x( x+1) + 3( x+1)= 0

( x+1)(2x+3)= 0 

ii. 5x²-17x+12= 0 factors of 60x²= (-5x)(-12x)

5x²-5x-12x+12=0

5x(x-1)-12(x-1)=0

(x-1)(5x-12)=0

Type 2

By multiplying the first and third terms and answer is negative then factors are dependent on middle term

If the middle term is of negative then greater factor is of negative

If the middle term is of positive then greater factor is of posive.

Examples: (i)10x²- 13x- 3= 0 factors of - 30x²= (-15x)(2x)

10x²-15x +2x-3=0 

5x( 2x-3)+1(2x-3)=0

(2x-3)(5x+1)=0

(ii) x²+x-132=0

x²+12x-11x-132=0 

x(x+12)-11(x+12)=0

(x+12)(x-11)=0

By using completing square

x²+2x+1=0

x²+2x= -1

(x)²+2(x)(1)= -1

Adding 1² on both sides

(x)²+2(x)(1)+(1)²= -1+(1)²

(x+1)²= -1+1

(x+1)²=0

Taking square root on both sides

( x+1)=0 

x=-1

Roots are real and same.

By Quadratic Formula 

x= -b± whole square root of b²-4ac is whole divided by 2a

4x²+12x+5=0 

By comparing with general equation

a=4, b=12 and c=5

x= -12± whole sqaure root of 144-80 is whole divided by 8

x= -12± whole square root of 64 and is whole divided by 8

x= -12± 8 and is divided by 8

x= -12+8 is divided by 8 and x= -12-8 is divided by 8

x= -4/8 and x= -20/8

x= -1/2 and x= -5/2

Note: Quadratic formula can be derived by using completing square.