Quadratic Equations

Quadratic equations are any of those equations which exhibit second degree. They can be written in a standard format as: mx2 + nx + q = 0. Here, m, n, and q are the numerical coefficients. The value of m should not be zero because that will convert the above equation into a linear one. This will happen due to the absence of the squared term i.e., mx2. We have to determine the value of x in the above equation to reach the desired solution. The unknown x terms are also called the roots of a quadratic equation. Quadratic equations are bound to have two solutions that can either be real or complex.

An alternative format to represent the quadratic equations is through the equivalent equations.

For example, we can consider the equation

mx2 + nx + q = m (x – t) (x – u) = 0

Here t and u are the two real solutions of x or the roots of the equation. Since 2000 B. C. we can express solutions of mathematical problems in terms of quadratic equation formula.

Characteristics of the roots of quadratic equation

Quadratic equations possibly have:

• Two roots that may be different and real numbers. 
• Two complex solutions or roots. 
• Double root or one real root. 

Let us consider one quadratic equation 6x² + 11x – 35 = 0. Here the roots are 5/3 and – 7/2. We have implemented the quadratic formula – – n ± √ (n2 – 4mq) to determine the roots. Here, a = 6, b = 11 and c = – 35. From the answers that we receive from the above equation, we can conclude that quadratic equations have different roots than are real numbers. To understand the nature of each root we have to take the help of the discriminant (n2 – 4mq).

If the value of the discriminant becomes greater than zero then we have two different roots which are two real numbers, i.e., they can be expanded to infinite decimal places. 

If the discriminant appears to be negative, we have to find out the square root of the negative integer as per the quadratic equation formula. Therefore, such quadratic equations will have complex roots. 

When the value of (n2 – 4mq) becomes 0, the quadratic equation formula changes into – n / 2m. In this case, we evaluate only one solution as the double root. 

How to solve Quadratic Equations? 

In the former section of the article, we discussed the nature of possible roots in a quadratic equation. Now let us focus on several methods that are implemented to find the solution of problems that involve quadratic equations. In several problems, we will notice that the equations are not in their quadratic standard format. We cannot proceed with any of the following techniques unless the quadratic equation is in its fundamental form. 

We will take the quadratic equation x2 + 2x – 3 = 0

• Factoring technique: Break the left-hand side of the equation into two factors. 

x2 + 2x – 3 = 0

⇒ x2 + 3x – x – 3 = 0

⇒ x (x + 3) – 1 (x + 3) = 0

⇒ (x + 3) (x – 1) = 0

As the product of the two factors is null, both of them should be zero. Therefore, x = – 3, 1. These are the roots of the equation. 

• Standard quadratic equation formula: 

When we compare the given quadratic problem to the standard format mx2 + nx + q = 0, we get m = 1, n = 2 and q = – 3

Quadratic equation formula:

– n ± √(n2 – 4mq)/ 2q

= – 2 ± √(22 – 4. 1. (-3)) / 2. 1

= – 2 ± √(4 + 12) / 2

= – 2 ± √16 / 2

= – 2 ± 4 / 2

The first solution of the quadratic equation is

(– 2 + 4) / 2

= 1

The second root of x is ( – 2 – 4) /2 = – 3

• Squaring method: We will take the equation  (x2 – 2x – 3) = 0

x2 – 2x – 3 = 0

⇒ x2 – 2x + 1 – 4 = 0

⇒ (x – 1)2 – 4 = 0

Let us add 4 on both sides. Then we get

⇒ (x – 1)2 = 4

⇒ (x – 1) = ± 2

Therefore, one root is (1 + 2) = 3 and the other one is (1 – 2) = – 1.

• The graph is used to manually mark the quadratic equation roots as x intercepts on the xy plane. Quadratic equations having complex roots cannot be solved using a graphical representation. Thus, it is always recommended to solve any quadratic equation using the quadratic equation formula. 

Conclusion

Quadratic equations have two roots which can be determined by factoring or through the quadratic equation formula. These equations are univariate as they are polynomials with only one variable. To qualify as a quadratic equation the variable must have the power of 2.