What is a Quadratic Equation?

When it comes to the concept of polynomial equations, quadratic equations can be said to be a special case. What does solving a quadratic equation mean? We will understand the quadratics and their types once we are familiar with the polynomial equations and their types.

In simple math's, a polynomial can be termed as an expression in algebra with non-negative integer exponents. Various coefficients and variables are also involved in this expression.

In general form, a single variable polynomial can be expressed with both constants and variables in it. With the increase in the number of terms, both the index of the constants and the power of the variables decreases.

Upon multiple forms, we have seen that the variables in a polynomial can only have non-negative integer numbers as their powers. Also, a polynomial is denoted by the highest power present on it. This is known as the degree of a polynomial. The word quadratic means two. Thus, they have degree two.

Apart from a quadratic equation, a zero-degree polynomial is known as a constant or a zero-order polynomial. If a polynomial equation has a degree of 1, then it is called a linear equation. For a degree of 3 and 4, it is known as cubic equation and bi-quadratic equation respectively.

How did Quadratic Formula Come Up?

In mathematics, the field of dealing with unknown variables and constants mainly lies in the algebra section. However, the quadratic equation has its origins in geometry. Renowned Greek mathematician Euclid came up with the quadratics for solving polynomials of degree two. However, in the same regard, the quadratic equation was also solved in a general form. This method was discovered by Brahmagupta, an ancient Indian mathematician. Around 1500 years ago, he came up with the solution technique. 500 years later, another ancient Indian mathematician Sridhar Acharya derived the well-known formula for solving these equations.

Where do you Use Quadratics in Real Life?

In real life, it has an important role. It is used when the profit of a product is calculated, or in estimating an object’s speed, it can come in handy.

How can we Express it in its Standard Form?

In general terms, we can call any polynomial $P\left(x\right)$ as a quadratic polynomial which has 2 degree and can be expressed in an equation as-

$P\left(x\right)=0$

This form can be arranged in many ways. We define a quadratic in its standard form where the individual terms are sorted and arranged based on their powers. The individual terms get arranged in decreasing order of their powers and equated to zero. This is the standard form. It has been expressed below-

Here, the terms p, q, and r can be said to be the coefficients. The terms$x^2$, x and $x^0$ have p, q, and r as coefficients respectively. The term ‘r’ can also be called a constant. In the above equation, the unknown quantity is noted by x. Hence, x can be said as the variable. These three terms constitute a general quadratic equation.

What is the Degree of a Polynomial?

In normal terms, a degree of a polynomial can be said to be the highest power that is present in a particular expression.

When it comes to the degree of a polynomial, the value holds significance. This is because the degree also states how many solutions it can have.

How many Solutions can a Quadratic Equation have?

Since it has a degree 2, it will have 2 solutions. If a polynomial has a degree of n, then there will be at most n possible solutions for it.

The degree can also be represented graphically. In a graph, the polynomial will touch the x-axis. The number of times the x-axis is cut by the graph of a polynomial denotes its number of solutions. If it does not touch the x-axis, then the polynomial doesn’t have any solutions.

Why is Quadratic Equation Graphed as a Parabola?

For a quadratic equation represented on a graph, there will be two points where the x-axis will get cut. For $y=p{x}^2+qx+r$, if p is positive, then the parabola opens upwards. If it is negative, then the parabola opens downwards. The x-intercepts are found by substituting the value of y as 0. This gives $p{x}^2+qx+r=0$.

How many Methods to Solve Quadratic Equations?

When we try to solve a quadratic equation, we usually try to satisfy the equation by finding the proper variable values.

We usually solve it in two ways. They are as follows-

• Factoring Method
• Quadratic formula

We will discuss both of these methods below.

Factoring Method

As the name suggests, the factoring method for solving quadratic equations is based on factors. The factoring theorem suggests that any value that can solve the equation can be termed as its factor. For instance, if a function $f\left(x\right)$ gives remainder equal to zero for $x=h$, then the term $\left(x-h\right)$ can be said to be a factor of the polynomial.

In the factoring method, we try to evaluate the equation and find out the factors that can solve the equation. In this process, the remaining function also gets simplified. Whenever we take a factor out from a quadratic equation, the remaining part becomes a linear equation, whose factor can be easily found out. Since a quadratic equation has a degree of 2, there can only be two factors.

When do you Solve Quadratic Equations by Factoring?

It is done when the terms in the equation can be grouped together and written as common factors.

Quadratic Formula

By the quadratic formula, the roots of the equations are calculated by the method completing the squares. If we have a standard polynomial of degree 2 like-

$a{x}^2+bx+c=0$

In the terms ax2, bx, and c, ‘a’, ‘b’, and ‘c’ all belong in the real number domain, and ‘a’ cannot be zero. In ax2, a is the coefficient of $x^2$, in bx, the coefficient of x is b.

Then the two roots of the equations are solved by using the following quadratic formula-

$x=\frac{-b+\sqrt{b^2-4ac}}{2a}$ or $x=\frac{-b-\sqrt{b^2-4ac}}{2a}$

Here, $\left(b^2-4ac\right)$ is known as the discriminant. In the formula, we take the square root of the discriminant. It is denoted by D or delta. The nature of the roots can be determined by the value of the discriminant. They can be complex roots or real solutions.

How to Solve the Quadratic Formula Step by Step?

Step 1) To use the quadratic formula, the equation must be equal to 0.

Step 2) Find out the values of ‘a’, ‘b’ and ‘c’ and substitute them in the quadratic formula to find the solution.

The Nature of Roots

The discriminant determines what the nature of the roots will be. Based on this value, we can conclude whether the pair of roots in a quadratic equation are real and equal, real and distinct, or complex.

When does a Quadratic Equation have Equal Roots?

When $D=0$ both the roots of the quadratic equation are equal in value. The values are also real, and when represented in a graph, the curve meets the x-axis at one point only.

When $D>0$ we get real and distinct roots. In a graph, the curve cuts the x-axis at two different points.

When $D<0$ the roots can be said to be complex, non-real, or imaginary in nature. Hence, there are no points of contact of the curve with the x-axis graphically.

Special Cases

It is to note that when the coefficients come out to be rational, but the roots are irrational, then, the roots are conjugate of each other.

If there are imaginary coefficients, then the roots are complex conjugate of each other.

Practice Problem

Example 1

Solve the equation $x^2+10x+9=0$ by factoring

Solution :

The above equation can be factored as follows

Thus, -1 and -9 are the roots of the given equation.

Example 2

Solve the quadratic equation x²+4x-21=0 using the quadratic formula.

Solution: Find using the quadratic formula.

Compare x2+4x-21=0 with $a{x}^2+bx+c=0$:

In the place of ax2 we have x2, in the place of bx we have 4x, and for ‘c’ we have $-21$

Here, $a=1$, $b=4$ and $c=-21$.

Substitute the values in the quadratic formula.

Rewrite the above equation as follows.

Formula

The quadratic formula for the solution of equations of the form $a{x}^2+bx+c=0$ is $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, where a, b and c are real numbers such that a is non-zero.

Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for:   

• Bachelor of Science in Mathematics       
• Master of Science in Mathematics