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Quadratic Equations

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Last updated date: 16th May 2024
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What are Quadratic Equations?

A quadratic equation is a polynomial equation where the highest power attached to a variable is of order 2. As the highest power of the variable attached to the polynomial equation is two, it means that at least one term in the equation exists, which is squared. Because of this, the equation is called “quad”.


A quadratic equation can be expressed in the general form of ax2+bx+c=0, where a, b and c are numerical coefficients or constants, and the value of x is variable. One fundamental rule of a quadratic equation is that the value of the first constant never can be zero


These equations make up a significant part that is necessary to solve several kinds of complicated mathematical problems. The practical use of quadratic equations is extensively seen while calculating the dimensions of parabola, the speed, and other dimensions of projectile motion involving athletics and sports. 


Solutions of a Quadratic Equation

There are two fundamental methods to find the roots of a quadratic equation. They are 

  • The Standard Equation Method

  • The Factorisation method.


According to the Standard equation method, the roots of a quadratic equation can be found by the


x = \[\frac{-b \pm \sqrt{b^{2} - 4(a)(c)}}{2a}\]


This equation is also known as the Sridharacharya Formula.


While in the Factorisation method, the constant “b” is broken down into such quantities so that the entire equation becomes a perfect factor where x can be taken as the common factor in ax2+bx.


Basic Concepts

There are several ways in which you can calculate quadratic equations. However, two of them are the most useful and accessible. 


Standard Equation

Using this method, you can solve any quadratic equation. Suppose the equation is ax² + bx + c = 0, hence the value of x will be x = \[\frac{-b \pm \sqrt{b^{2} - 4(a)(c)}}{2a}\]


It is also known as the Sridharacharya formula. 


Employing this technique, you will get two types of value. One of these will be positive, and the other one will be negative. 


Factorising Method

This method is in quadratic equation class 10 syllabus and therefore essential for you to learn about this in detail. 


In this method, you obtain the solution factoring quadratic equation terms. However, for this, the equation has to be eligible for factoring. This technique is easier than others. 


Consider this example of a quadratic equation and find the solution. 


 x² -5x + 6 = 0


The equation is the standard form quadratic equation. To find the solution of it, first you have to consider two terms that are b and c. In this case, b = -5 and c = 6. Now you have to find the product of which two numbers will be 6. Also, the sum of the two numbers has to be -5.


For that, find the factors of 6, which can be 1, 2, 3, and 6. Thereby, you will see that the product of 2 and 3 is 6 whereas the sum of 2 and 3 is 5. Since we have to find -5, we have to take -2 and -3, only then the sum will be -5 and the product will be 6.

Therefore, two factors of the equation are (x + 2) and (x + 3).


Quadratic Equation Problems

Following is the quadratic equation with solution

3x2 - x = 10

 3x2 - x - 10 = 0

 3x2 - 6x + 5x - 10 = 0

 3x (x - 2) + 5 (x - 2) =0

 (x - 2)(3x + 5) = 0

Therefore, x - 2 = 0, x = 2

And when 3x + 5 = 0; 3x = -5 or; x = -5/3

 Thus, x= -5/3, 2


Conclusion

If you want to solve quadratic equations online, tune in to our website. Along with the notes, you can get access to the quadratic equation NCERT solution for grade 10. 

FAQs on Quadratic Equations

1. What is the use of Quadratic Equations?

Quadratic equations are among the most widely used mathematical equations which have a widespread application in the field of coordinate geometry and also in applied physics. A parabola is a type of basic curve that is generally of the form of y=ax2+c, where a and c are constant. This is a very important formula from the perspective of Newton Physics as the projectile motion is also a parabola and any object which is thrown with some velocity in a horizontal direction, under the influence of gravity undergoes projectile motion.

2. How can we find out if a quadratic equation has real or imaginary roots?

To find out what types of roots a quadratic equation has, we use the Sridharacharya Formula, as the formula has a term b2-4ac which is under roots. If this term is positive, that is b2-4ac has a positive value then the quadratic equation has two real and distinct roots, if the term b2-4ac is 0 then the quadratic equation has just one real root and if the term b2-4ac is negative then the quadratic equation has imaginary roots. 

3. What are imaginary roots to a quadratic equation?

In the study of algebra, you will often find that there are certain quadratic equations and their solutions contain the negative square roots. Such square roots are known as imaginary numbers, and the roots are called complex roots. However, these complex roots can be expressed as a ± b.


The Imaginary numbers in quadratic equations occur when the value of the fundamental part of the quadratic formula is negative. In such a situation, the equations do not contain zeros or roots in real number sets. 

4. What is the historical context of Quadratic equations?

It is claimed that the Babylonians first solved quadratic equations about 400 B.C. however, they did not have the idea of an equation, and they used to employ an algorithm to solve mathematical problems. It is also believed that later in 300 B.C. Euclid and Pythagoras developed an approach to solve these equations. They used to follow this method to find the length of an object, which is now presumed as the root of a quadratic equation. From the Indian perspective, Brahmagupta also gave a modern method to solve this equation. Therefore, it can be said that the contribution of all these illustrious minds gave birth to the modern version of a quadratic equation.

5. How to Find Roots Quadratic Equation Using Factoring?

Using the following steps, you can easily solve quadratic equation by factoring. 

First of all, keep all the terms on the left side of the equal sign, leaving zero on the right side. Then, set each factor= 0. Start solving individual equation. Once you get the answer, put the values in the equation for crosschecking. 


For example, 

x² – 6 x = 16 

It becomes x² – 6 x – 16 = 0

Factors are.

(x – 8)(x + 2) = 0

Setting each factor =zero, 

We get, x - 8 = 0; x + 2 = 0

Therefore, x = 8; x = -2

To check, 

8² - 6(8) = 16 or (-2)² -6(-2) = 16

64 - 48 = 16    4 + 12 = 16

16 = 16    16 = 16

Answer: 8 and –2 both these values are solutions of the original equation.

6. Who Invented Quadratic Equation?

Often it is claimed that Babylonians first solved quadratic equations about 400 B.C. however, they did not have the idea of an equation, and instead, they used to employ an algorithm to solve mathematical problems. 


Also, it is believed that later in 300 B.C. Euclid and Pythagoras developed an approach to solve these equations. They used to follow this method to find the length of an object, that is now presumed as the root of a quadratic equation. 


Brahmagupta in India also gave a modern method to solve this equation. Therefore, it can be said that the contribution of all these legends gave birth to the modern version of quadratic equation.

7. What is Quadratic Formula and Where it is Used?

Quadratic equation formula is a method to solve quadratic equations. However, there are other methods as well to solve such kind of equations. One of the significant derivations of this formula is completing square formula. It can be used in the following steps.


Firstly, you have to divide each side by a. Then rearrange. Next (b/2a)² has to be added to each side. Now, rearrange all the terms on the right side to get a common denominator. Then, square root both sides and finally isolate x.