# Formation of Quadratic Equation in One Variable – Definition, Formula, Tricks, Examples | How to Solve Quadratic Equation in One Variable?

In this article, you will learn about the formation of Quadratic Equations in one variable. An equation in which the variable varies to a degree of two is a quadratic equation. In other words, an equation in which the variable has the maximum degree of two is a quadratic one. Quadratic means related to a square. A mathematical statement in which two expressions on both the left-hand side and the right-hand side of an equality symbol are equal is an equation. Algebra makes it easier to solve real-world situations. The formation of Quadratic Equations can be done by using variables, constants, and an equality sign.

On this page, we will discuss the definition, the relationship between roots and the co-efficient of a quadratic equation, and formation of quadratic equation in one variable solved examples, and so on. you have a step-by-step solution for all the problems provided and get a good knowledge of this concept. Find various 10th Grade Math concepts similar to Quadratic Equations all arranged efficiently.

### What is the Formation of Quadratic Equation?

It is defined as a Quadratic equation having a degree of 2 that is written in the form of ax^2+bx+c = 0, where a, b,câˆˆR and aâ‰ 0. Every quadratic equation has a corresponding quadratic polynomial that you get by changing the “0” to an “f(x)”. Standard form for a quadratic polynomial is f(x)=ax^2+bx+c.

### How do you Solve Quadratic Equations with One Variable?

The solution of a quadratic equation is called the roots of the equation. Thus, x=Î± is a root of f(x)=0 if and only if f(Î±)=0. Also, by the factor theorem, (xâ€“Î±) is a factor of f(x) if and only if Î± is a root of the quadratic equation f(x). Therefore, a quadratic equation f(x)=0 has:

• either two distinct real roots.
• two equal real roots.
• no real roots.

If a quadratic equation has real roots, it can have only two of them. As a result, the quadratic equation can have no more than two real roots. With a term and with the variable being increased to the second power, the quadratic equations vary from the linear equations. Because of adding, subtracting, multiplying, and dividing the terms will not isolate the variable, so we can utilize different strategies to solve quadratic equations than linear equations.

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Relation between Roots and the Coefficients of a Quadratic Equation
If Î±,Î² are roots of the quadratic equation ax2+bx+c=0, where aâ‰ 0, then:
1. The sum of roots = Î±+Î² =âˆ’Coefficient of x/Coefficient of x^2= âˆ’b/a
2. The product of roots = Î±Î² = Constant Term/ Coefficient of x^2= c/a

### Formation of Quadratic Equation in One Variable Examples

Problem 1: The product of two consecutive even integers is 168. Form the quadratic equation to find the integers if x denotes the smaller integer. Find the value of the integers?
Solution:
Given that,
The first even integer is x.
Next, the second consecutive even integer will be x+2.
The given product is 168, the equation will be as follows:
x(x+2)=168
x2+2x=168
x2+2xâ€“168=0
Now, we need to find the discriminant D value that is D =b^2â€“4ac.
D = 4+4Ã—168=676
Since D>0, the two real and distinct roots are exist.
x = â€“bÂ±âˆšD/2a
x = â€“2 Â± âˆš676/2
x = â€“2Â±26/2
Both the roots satisfy the condition of being even integers.
Hence, the first possibility is two consecutive positive integers are 12 and 14.
The second possibility is two consecutive negative integers are âˆ’12 and âˆ’14.

Problem 2: The perimeter of a rectangle is 20 cm and its area is 24 cm2. Formulate the quadratic equation to determine the length and breadth of the plot.

Solution:
As given in the question, the values are
The length of the rectangle is x cm.
The breadth of the rectangle is (20âˆ’2x/2) cm = (10âˆ’x) cm.
Next, the area of the rectangle is 24 cm. sq.
Now, we can formulate the quadratic equation,
Therefore, the equation is x(10âˆ’x) = 24
10xâˆ’x^2 = 24
So, the quadratic equation is x2âˆ’10x+24=0. This is the required quadratic equation.

Problem 3: Raju works 3 hours more to do a piece of work than john. They together complete the hour in 2 hours. Find the quadratic equation?

Solution:
In the given question, the data is
The john completes a piece of work in x hours.
The Raju completes the same work in (x + 3) hours.
Therefore, in one hour john completed 1/x part of the work
In 1 hour Raju completed 1/(x + 3) part of the work.
So, in 1 hour they togetherly complete work is 1/x + 1/(x + 3) of the part.
According to the problem we get,
1/x + 1/(x + 3) = Â½ ……………….. (i)
Or
Let Raju complete a piece of work in x hours
The john completes the same work in (x – 3) hours.
Therefore, in one hour john completed 1/(x – 3) part of the work.
In 1 hour Raju completed 1/x part of the work.
Hence, in 1 hour they togetherly complete 1/(x- 3) + 1/x of the part.
According to the problem we get,
1/(x – 3) + 1/x = Â½ ……………….. (ii)
Both the equation (i) and the equation (ii) are quadratic equations.

Problem 4: Find the quadratic equation whose solution set is {2/3,âˆ’3}.

Solution:Â
As given in the question, the solution set is {2/3, -3}.
Now, we need to find the quadratic equation.
The Roots are 2/3 and -3.
The Sum of Roots are (2/3)-3 = (2-9)/3 = -7/3.
Next, the product of roots are (2/3)(-3) = -2.
So, the quadratic equation whose roots are 2/3 and -3 is
x^2âˆ’(sum of roots)x + product of roots = 0.
x^2âˆ’(âˆ’7/3)x+(âˆ’2) = 0.
x^2+7/3x -2 = 0.
3x^2+7xâˆ’6=0.
Thus, the quadratic equation is 3x^2+7xâˆ’6=0.

### FAQs on Formation of Quadratic Equation in One variable

1. What is an example of a quadratic equation in one variable?
The example of the standard form of a quadratic equation is (axÂ² + bx + c = 0) which include 6xÂ² + 11x – 35 = 0.

2. How do you make a quadratic equation?
A quadratic equation can be formed when its roots are given or the sum and product of the roots are given. A quadratic equation whose roots are Î± and Î² is, (xâ€“Î±)(xâ€“Î²) = 0 or,
x2â€“(Î±+Î²)x + Î±Î²=0 or,
x2â€“(sum of roots)x + (product of roots)Â  = 0.

3. What are the types of equations?
The degree of an equation is the highest power of variables in a term in an equation. Based on the degree of the equation, the equations can be classified as:
1. Linear equation, whose degree is 1.
2. Quadratic equation, whose degree is 2.
3. Cubic equation, whose degree is 2.