Quadratic Equation in Real and Complex No System

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The Quadratic equation is one of the most popular pre-calculus mathematics topics that students in high Scholl will have to encounter. Unlike linear algebra, that deal with linear equations (where the x can only have a power up to 1), the quadratic equation deals with the variable X up to power 2.

The three ways of computing the Quadratic equation are Factorization, The Quadratic Formula, and the Completion of Squares Method. All three methods will lead to the root solution of the quadratic equation. The Quadratic Equation is considered the Alpha formula. This s because quadratic equation solutions that may be difficult to reach using factorization or completing the Squares method can be easily derived with it.

This article will consider the quadratic equation using all the methods to determine the real number and complex number roots.

The Quadratic Equation

A quadratic equation must have three real known numbers. The first number is usually labeled as ‘a’ and is the coefficient of X2. The other number which is labeled as b is the coefficient of just X, while the last number c is the constant. There are three major ways to compute the quadratic equations to get a value or Values for X. The values of X are called the solution of equations, which means that there are the solutions that will fit the Quadratic equation.

The Quadratic Equation Formula

The Quadratic Equation is any type of equation whose standard form is

ax2 + bx + c = 0 …….. eqn (1)

where a, b & c are all real numbers.

The aim of the quadratic equation is to find values for x’s such that when they are substituted into the left hand of the equation, the outcome would indeed equal 0 on the right-hand side.

The General formula for the Quadratic equation is given as;

the a, b, and c in the above formula will represent the values that the same values represent in the general quadratic equation in Eqn (1)

When solving the solutions of a quadratic equation, It is important to know that the highest solution that can be solved is 2. In some instances, x may only have one value. Usually, though x always gives two values.

Apart from the General quadratic formula, Factoring and Completing the Squares are two other methods for getting the solutions to the quadratic equation.

How to Solve a Quadratic Equation?

The Factoring Method

The factoring or factorization method is the most popular way of solving the quadratic equation. Below are the steps to solve it.

Step 1: identify the a, b c of the quadratic equation

Step 2: multiply a and c together to form a new coefficient for c. This will lead to a new quadratic equation.

Step 3: Expand the equation. To do this, find two numbers such that when they are added together, it will give ‘b’, and when they are multiplied, it will give c

Step 4: Continue to factor until the value or values for X is found.

Completing the Square Method

It is important to understand that this method is preferable when dealing with ‘a’ = 1. If a = any other value that is not 1, then it is best to use the factorization method or Quadratic formula.

Step 1: Identify a, b and c

Step 2: move the ‘c’, which is the constant C to the other side of the equation

Step 3: Divide the ‘b’ by 2. Also, the answer reached will be squared and added to the equation from both sides.

Step 4: Now, working with the derived equation after step 3 completion,Pick ‘a’, also pick the sign after it, and pick ‘c’, then square up

Step 5: Take the square root of both sides

Step 6: Find the value of x

The General Quadratic Formula

Step 1: write out the general formula for finding the solutions and identify the a, b and c from the given equation

Step 2: solve for the two x’s

Calculating Quadratic Equations Using all Three formulas

Find the roots of the quadratic equation x2 – 2x – 8 = 0

Example 1

Step 1: identify the a, b c of the quadratic equation

a = 1, b = – 2 and c = – 8

Step 2: multiply a and c together to form a new coefficient for c. This will lead to a new quadratic equation.

Clearly the initial quadratic equation is x2 – 2x – 8 = 0

Since a = 1 and c = – 8, multiplying them to form a new c will still leave it equal to the same c

So the new equation will still be

x2 – 2x – 8 = 0

Step 3: Expand the equation. To do this, find two numbers such that when they are added together, it will give ‘b’, and when they are multiplied, it will give c. These two numbers will replace bx

From the equation, b = – 2, and c = – 8,

2 and – 4, when added, will give -2 and when multiplied, will give – 8

Also, these two numbers will replace bx. So;

 x2 – 2x – 8 = 0

x2 + (2 – 4)x – 8 = 0

X2 + 2x – 4x – 8 = 0

Step 4: Continue to factor until the value or values for X is found.

(x-4)(x + 2) = 0

x + 2 = 0, x – 4 = 0

x = – 2, x = 4

Example 2

Solve the Quadratic Equation X2 – 2x -15 = 0 by using the Complete Squares Method

Solution

Step 1: Identify a, b and c

From the equation x2 – 2x -15 = 0

a = 1, b = -2 and c = -15

Step 2: move the ‘c’ which is the constant to the other side of the equation

x2 – 2x = 15

Step 3: Divide the ‘b’ by 2. Also, the answer reached will be squared and added to the equation from both sides.

That is,  

x2 – 2x/2 = 15

According to the step, the square of the divide b will be added to both side of the equation so

x2 – 2x/2 + (-2/2)2 = 15 + (-2/2)2

x2 – x + (– 1)2 = 15 – 12

x2 – x + (– 1)2 = 14

Step 4: Now, working with the derived equation after step 3 completion, write down x, then pick the sign after ax2 and pick ‘c’. Join them together and square up to get a new left-hand side

Clearly; the derived equation after step 3 is x2 – x – 12 = 14

the sign after x2 is the minus sign (-), while its c = (-1)

Following step 4; now we write down x, the minus sign and -1 and square up. This will give us

(x – (-1))2 = 14

(x +1)2 = 14

Step 5: Take the square root of both sides

Step 6: Find the value of x

Base on the fact that the equation a go two ways because of he + and – sign, we will have

X + 1 = 3.742

and

X + 1 = – 3.742

Solving for the first

X + 1 = 3.742

X = 3.7412 – 1

X = 2.741

Solving for the second

X + 1 = – 3.742

X = – 3.742 – 1

X = – 4.741

Therefore; x = 2.741, -4.741

Example 3

Find the solution of the equation using the quadratic equation.

2x2 + 9x + 7 = 0

Solution

Step 1: write out the general formula for finding the solutions and identify the a, b and c from the given equation

The quadratic equation is given as

The Complex Number System

A value is considered to be complex or in the complex number system if it is given as a + bi where a is a and b are real numbers

and

for example; 6 + 7i is a complex number.

A quadratic equation that is said to be in a complex name if its roots are given in the form of  a + bi

Example 4

Solve x2 – 4x + 8 = 0

Solution

We, will, use the Quadratic Equation

This means that the solution will be

Clearly, The Roots Are Complex Numbers, and That Is What It Means to have a complex plane

Conclusion

This article has fully considered calculating the quadratic equation in both the real number and complex number systems. More often than not, you will need to make many computations of the quadratic equation in real numbers. However, there are times where your calculation may end up in a complex root. When that happens, then you will be calculating in a complex plane.

Quadratic Equation in Real and Complex No System

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