Introduction for Algebra Quadratic Equations

Introduction for Algebra Quadratic Equations:

Algebra :It is a branch of mathematics that includes terms,equations, polynomials and algebraic structures.The most basic form of algebra is elementary algebra.The terms that are related with algebra are
*Variables these are alphabetical characters that are used for assigning the value.generally we use x,y,z as variables.
*Terms:It is a single number or a variable and numbers multiplied together.Example 2x-5=3 here 2x,5,3 are called terms
*Constant :It is fixed value and does not contain any variable.Example the constant term in 3x+2 is 2.
In algebra polynomials is an expression that contains one or more variables and constants using operations(addition,subtraction and multiplication).Quadratic equation is an example of polynomials.

Quadratic Equation:
   It is an equation that contains a second Degree term. The quadratic equation is of the form ax^2+bx+c=0 where a, b, c are the coefficients (numbers) and X is the variable .The Variable X is the solution (or we can say it is a root) if it satisfies the given equation ax^2+bx+c=0.
Some Examples of quadratic equation are given as
x²+3x+4=0        3x²+9x-2=0            3x²=12
Quadratic equations can be solved by:

• Using Simple Factoring

• Using Quadratic Formula

• Using the square root method.

• Graphing.

• Completing the square method

Solving Quadratic Equations by Method of Factorization:

Step 1: Write the equation  in standard form, ax^2+bx+c=0,where a is not equal to zero, if needed.
Example: X²+7x-8=0

Step2: Find the product ac .Determine all possible factors of the first and third term ac= -8 all possible factors of 8 are (2 * 4) and (1 * 8).

Step3: Choose the factors that will add to give the middle term. Addition and subtraction sign tell you whether the term should be positive or negative. Since ac = - 8 we will need a positive and negative factor. Since the middle term, is positive, the positive factor of 8 will need to be larger than the negative factor. For instance, (-8*1) will give -8 but result is -7.So we take 1 as negative factor and 8 as positive.

Step 4: The final factors will be in the form (x+ ) (x+ ) 
This trinomial will have the factors (x-1) (x+8). 

Step5. Use the Zero product Principle 
To solve for X set each factor= 0
x-1= 0 that given x=1 and x+8= gives x= - 8

Quadratic Formula to Solve Quadratic Equation in Algebra

Quadratic formula is given as

      The General Quadratic Formula to find the roots of a quadratic equation is,

                        x = `((-b+-sqrt(b^2-4ac))/(2a))`

Where,

            a = coefficient of x²

            b= coefficient of x

            c= constant

            b²-4ac = discriminant

This formula generally gives 2 solutions. The Plus and minus sign shows that there are 2 possible 
solutions.

Deriving the Quadratic formula:

It is obtained by solving the general quadratic equation

  aX²+bX+c=0

Divide each side of equation by a

  X²+(b/a)X+c/a=0

Subtract c/a from each side of the equation

 X²+(b/a)X=-c/a

To complete the square add (b/2a)² to each side of the equation

  X² + (b/a)X + (b/2a)²=-c/a + (b/2a)²

  (X+b/2a)²= -c/a+ (b²/4a²)

Taking LCD for the right side of equation

(X+b/2a)²= (-4ac/4a²)+ (b²/4a²)

(X + b/2a)²= (-4ac +b²)/4a²

Take the square root of each side of the equation.

  X+ (b/2a) =  ±√(b²-4ac) / 2a

   X = `((-b+-sqrt(b^2-4ac))/(2a))`

When a quadratic equation is in standard form ax^2+bx+c=0 , the expression, b^2-4ac, that is found under the square root part of the quadratic formula is called the discriminant.

Discriminant formula 
    

 d=b²-4ac 
Kinds of solutions for ax²+bc+c=0
a)If Discriminant d is greater than zero(means it is positive) b²-4ac>0 :The equation has two solutions both of which are real numbers. Or we can say it has 2 distinct real solutions

b)If Discriminant is zero  b²-4ac=0 : The equation has only one real solution

X=-b/2a
c) If Discriminant Is less than zero( Discriminant is negative)b²-4ac<0 span="span">  : No solutions are defined.
There are two distinct (non-real) complex roots, which are complex conjugates of each other:

  (-b + i √4a c-b²)/2a       and       (-b- i √4ac-b²)/ 2a
where i =`sqrt(-1)` 

Nature of Roots of a Quadratic Equation

The quantity D = b²- 4ac is known as the discriminant of the quadratic equation whose roots are given by:
        α =  `(-b + sqrt(b^2-4ac))/(2a)`     and
       β = `(-b- sqrt(b^2-4ac))/(2a)` 
The nature of roots is as given below:

1. The roots are real and distinct iff D>0.
2. The roots are real and equal iff D=0.
3. The roots are complex with non-zero imaginary part iff D<0 .="." span="span">
4. The roots are rational iff   a,b,c are rational and D is a perfect square.
5. The roots are of the form p+sqrt(q) (p,q belongs to Q) iff a,b,c are rational and d is not a perfect square.
6. If a=1, b,c belongs to I and the roots are rational number, then these roots must be integers.
7. If a quadratic equation in x has more than two roots, than it is an identity in x that is a = b = c = 0. 

Square Root Method to Solve a Quadratic Equation in Algebra

Steps to Solve Quadratic equation Using Square Root method

Step 1: Write the quadratic equation in the form A²=B if needed.
 Example X² = 16 

Step 2: Apply the square root method. 

 If A and B are algebraic expressions such that A² = B ,then

 A is equal to Square root B or A is equal to - Square root B

 Also it is written as A = plus minus root B

X²=16

X=√16  or X= -√16

There are 2 solutions

Step 3: Solve for the linear equation(s) set up in step 2.

  X=√16  that means  X=4

 X=-√16  that means X=-4

There are two solutions to above quadratic equation that is given as x = 4 and x = -4.