Quadratic equations with Real Coefficients

Quadratic Equation is an equation of the form ax2+bx+c = 0, meaning where the maximum degree of the variable X is 2. Here a,b and c are the coefficients of the variable x. Always remember that the value of a cannot be equal to zero for this equation to be called Quadratic, as the combined value of ax2 will become zero if a=0, and hence the resulting equation would not be a quadratic equation anymore. ax2+bx+c is a quadratic expression and when the quadratic expression is equated to zero(0), i.e., ax2+bx+c =0, the expression becomes a Quadratic Equation or Quadratic Polynomial.

In the polynomial, ax2+bx+c, the expressions ax2, bx and c are called the terms of that polynomial. Each term of a polynomial has a coefficient and a variable. Therefore, we can say that the algebraic expressions are of form (a constant)*(a variable). The value of constant remains the same throughout a particular situation, but the value of a variable can keep changing.

Quadratic Equation with Real Coefficients:

A quadratic equation with real coefficients is an expression of the form:

P(x) = anXn + an-1Xn-1 + …………… + a1X + a0

Where the numbers an, an-1, ……………, a1,a0  are the coefficients of the polynomial and will only be the real numbers. For the term anXn, coefficient an can never be equal to zero for the degree of X=2 for the given term. an is also called the leading coefficient.  And the degree of variable X is a non-negative integer and as the given polynomial is of the form Quadratic Polynomial, the highest degree of X will be 2.

An Example of a Quadratic Equation with a real coefficient can be as follows:

X2 -3X + 2 = 0

We can solve the given quadratic equations using either factor method, completing the square method, Sri Dharacharya formula or some other given methods.

Using Factor Method here, we will get:

(X-2) (X-1) = 0

X = 2,   X = 1

These values of X are called the Roots of the Quadratic Equation.

Classification of Polynomials:

Linear Polynomial:

When the highest degree of the given polynomial expression is 1, the polynomial/equation is said to be Linear Polynomial. For example:  7a+5

Quadratic Polynomial: 

When the highest degree of the given polynomial expression is 2, the polynomial/equation is said to be Quadratic Polynomial. For example: 2x2 + 10x + 5

Cubic Polynomial: 

When the highest degree of the given polynomial expression is 3, the polynomial/equation is said to be Cubic Polynomial. For example: 2x3 + 10x + 5

Biquadratic Polynomial:

When the highest degree of the given polynomial expression is 4, the polynomial/equation is said to be a Biquadratic Polynomial. For example: 2x4 + 10x3 + 5x2 + 48

Now, if we have a polynomial expression of degree greater than 4, then there is no specific name for those expressions. We can write them as:

For example: 2x9 + 10x3 + 5x2 + 48

The highest degree here is 9, hence it can be termed as a polynomial expression of degree 9.

Note:

(i) Degree of Polynomial is the highest power of variable x.

For example:  2x3 + 10x + 5 = 0

Here the degree of Polynomial will be 3, as the highest power in the given equation for the variable x is 3.

(ii) The degree of X can only be Non-negative Integers. Therefore, if the degree (power) of variable x is negative or not a whole no.(fraction), then the expression will not be Polynomial. 

For example:

10x-3 + 48 is not a polynomial as the variable x has a degree -3(negative), which is a negative integer.

Coefficients and their types in Quadratic Equation:

Coefficients are the values coming before the variables in any algebraic expression/equation. For example: In the given equation, 2x3 + 10x + 5, we have one variable named x with degree 3 and 1 respectively and three coefficients a, b and c.

Values of Coefficients will be as follows:

Coefficient of X3 = a = 2

Coefficient of X = b = 10

Constant value coefficient = c = 5

Real Coefficients:

Real Coefficients are those which are not imaginary. Real numbers are either rational or irrational. Rational no. is any number that can be written in the form p/q, where p and q are two integers. And irrational numbers are those that give non-repeating decimals that are non-terminating, meaning they continue forever with no pattern.

If the value of coefficients a, b and c are real, then the equation is called the Quadratic Equation over Real Coefficients.

Imaginary Coefficients:

Imaginary numbers are generally the roots of negative numbers. They are written as i (I in Italics basically). For eg : √ -1  = i

If the value of Coefficients a, b and c are imaginary, then the equation is called the Quadratic Equation over Imaginary Coefficients.

Conclusion:

Quadratic equations are studied under Algebraic Expressions in Mathematics. In this article, we discussed the detailed meaning of terms degree, expression, equation, coefficients, real and imaginary numbers and eventually Quadratic Equations in Real Coefficients. We learnt the classification of polynomials and possible types of coefficients and about various methods for solving a given quadratic equation. We hope this study material will give you a deep insight into all the basics required to understand the meaning of Quadratic Equation with Real Coefficients.