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In mathematics, linear inequalities are defined as equations that have inequality symbols; or linear inequalities in algebra are those inequalities that do not entirely satisfy an equation when represented in a mathematical expression.
An equation written in such a form, ax2+bx+c=0, is called a quadratic equation, where x represents a variable and a ≠ 0. Those values of ‘x’ that entirely satisfy the equation are called roots of the equation. In equations where the component of the quadratic equation does not make a complete square, this method is used to solve such equations.
Linear Equations and Inequalities
In algebra, equations are tools to express a mathematical problem or relationship between variables and constants. A linear equation is solved by calculating the value of the variables.
The Variables
The variable in an equation is defined as that component whose value may change or vary or is unknown. To understand variables, let us look at one example:
2x+9=0
In this equation, the variable is x because it is unknown. The equation will have different values for different values of x because x does not have a singular value, but a value that varies called a variable of the equation.
Normally, expression variables can be represented by x, y, z, a, b, c, …., etc.
An equation can also have more than one variable, for example, 3x+5y+45=0 or x+2y=10.
The Quadratic Equations
The quadratic equations are those equations that follow the standard equation, ax2+bx+c=0. The value of constant a can never be 0 in these equations.
The values of variable x that satisfy a quadratic equation are called the roots of the equation.
x2+2x+4=0, 3x2 +7x+21=0, 11x2-17x=57, etc are some of the examples of quadratic equations.
A quadratic equation can have the same, real, and even imaginary roots. The roots of a quadratic equation can be determined either by the factorization method or simply by using the quadratic formula.
The quadratic formula is given by,
[-b ± √(b2– 4ac)]/2a.
Completing the Square Method of Solving Quadratic Equations
We may have one of the components missing from a quadratic equation in some problems.
For example, if we have an equation such as 2x2+x=0, or 4x2+5=0, in such equations where we do not have values for some constant terms, we can not use the quadratic formula to determine the roots. The method used to determine the roots of such equations is called completing the square method.
Let us understand with an example:
Imagine a quadratic equation,
2x2-8= 0
In this equation, we can easily determine the value of x,
2x2=8
x2=4 .
x=2
However, this is only possible when we don’t have a value of ‘b’ in a quadratic equation.
Now, let us look at a different example,
Consider a quadratic equation, x2+Bx=0.
Where B is a constant value.
Since we don’t have the value of constant ‘c’, we can not use the square root method.
Therefore, we add a new constant D on both sides of the equation to get a perfect square equation.
x2+Bx+D= 0+D
Or, (x+D)2=0
From the above equation,
D=(B/2)2
The above method of determining roots of an equation that doesn’t have the constant term by making a perfect square trinomial is called Completing Square Method.
Solved Examples
Example 1: Solve the equation by using Completing the Square Method.
x2 + 4x =0
Solution: First we find the constant ‘d’ for the equation,
d = (b/2)2
d = (4/2)2
d =4,
Now, adding the value of d on both sides of the equation,
x2 + 4x + 4 = 0 + 4
Or, (x+2)2 = 4,
x+2 = ± √4
x+2 = ± 2
x = 0, or x = -4
Example 2: By using the completing the square method, find the roots of the quadratic equation.
x2+ 8x =0
Solution: First we find the value of the constant,
d = (b/2)2
d = (8/2)2
d = 16,
Adding the value of d on both sides of the equation,
x2+8x+16=0+16
(x+4)2 = 16
(x+4) = ± 4
Therefore, x = 0 and x = -8
Conclusion
Linear equations are those equations that represent a straight line upon plotting the values of the variables on a graph paper. The quadratic equations follow the standard equation given by ax2+bx+c=0, with a ≠0. In some quadratic equations, the value of one of the components of the standard equations is not given; to determine the roots of such an equation, completing the square method is followed. A constant is added on both sides of the equation to get a perfect square trinomial in completing the square method.
