In maths, the quadratic equation has one variable with the highest degree as two. We think all the students are familiar with the concept of solving quadratic equations. The term quadratic means square. Make use of the formulas to solve quadratic equation in one variable. Let us examine thoroughly about Formation of Quadratic Equations in One Variable with some examples here.
Formation of Quadratic Equation in One Variable
The solution of a quadratic or polynomial equation is called the roots of the equation. The equation with one variable x is called the formation of a quadratic equation in one variable. Quadratic equations are different from linear equations. We use different methods to solve quadratic equations such as factoring, squaring, and the quadratic formula.
A quadratic expression f(x) has
- Either two distinct real zeros
- It may have two equal real zeros
- It may have no real zeros
The quadratic equation f(x) = 0 has
- Either two distinct real zeros
- two equal real zeros
- no real zeros
How to Form Quadratic Equations in One Variable Step by Step?
Step 1: First utter the quadratic equation in the general form.
Step 2: Factorize the given quadratic equation.
Step 3: Apply the zero product property.
Step 4: Implement the set of each variable factor to 0.
Step 5: Finally solve the resulting equations.
Read More:
Formation of Quadratic Equation in One Variable Examples
Check out the below examples to know the formation of quadratic equations in a single variable. This may help you to overcome the difficulties in solving quadratic equations.
Example 1.
1. Which of the following is a quadratic equation?
(A) 2x² + 5x + 4 = 0
(B) 2x + 3 = 0
(C) y = x²
(D) x + 1 = 0
Solution:
Options A and C are the quadratic equations because it has the highest degree as 2.
Example 2.
The product of the two consecutive odd integers is 169. Form the equation.
Solution:
Given,
The product of the two consecutive odd integers is 169
Let the two consecutive odd integers be x, x + 2.
x(x + 2) = 169
x² + 2x = 169
x² + 2x – 169 = 0
Example 3.
The area of the rectangular playground is 48 square meters. The length of the rectangular playground is 8 meters more than its breadth.
Solution:
Given,
The area of the rectangular playground is 48 square meters.
The length of the rectangular playground is 8 meters more than its breadth.
Let the width of the rectangular playground is x.
Let the length be x + 8
Area of the rectangle = length × breadth
x × x + 8 = 48
x(x + 8) = 48
x² + 8x = 48
x² + 8x – 48 = 0
x² + 12x – 4x – 48 = 0
x (x + 12) -4(x + 12) = 0
(x + 12)(x – 4) = 0
x + 12 = 0
x = -12
x – 4 = 0
x = 4
The length cannot be negative so the solution is x = 4.
breadth = 4 meters
length = x + 8
= 4 + 8 = 12 meters
Thus the length of the rectangular playground is 12 meters.
Example 4.
Which of the following is not a quadratic equation.
(A) y = 4x²
(B) x + 4x + 2 = 0
(C) y = 2x
(D) x² + 2x + 1 = 0
Solution:
The options B and C are not a quadratic equation because it does not have the highest degree as 2.
(B) x + 4x + 2 = 0
5x + 2 = 0
It is a linear equation because it has the degree 1.
(C) y = 2x
It is a linear equation because it has the degree 1.
Example 5.
The product of two consecutive positive odd integers is 121. Form the equation.
Solution:
Given,
The product of two consecutive positive odd integers is 121.
Let the two consecutive odd integers be x, x + 2.
x(x + 2) = 121
x² + 2x = 121
x² + 2x – 121 = 0
FAQs on the Formation of Quadratic Equation in One Variable
1. What is the formula of quadratic equation in one variable?
The standard form of a quadratic equation is ax² + bx + c = 0.
x is a variable and a, b, c are constants with a ≠ 0.
The number of solutions that a quadratic equation has is two.
2. How do you extract quadratic equations with one variable?
First, we have to write the equation in the standard and then factorize the expression and apply the zero product property equal to zero. Solve the resulting linear equations.
3. What is the degree of the quadratic equation?
The quadratic equation is also called a polynomial equation. It is a second-degree polynomial equation with the highest power of 2.