In earlier classes, you have learned a variety of polynomials. For example, the quadratic polynomial of the form \(ax^2 + bx + c\), \(a \ne 0\) was one kind. We get a quadratic equation when we equate this polynomial to zero.
Let's see a conversation between two friends to understand the concept of the quadratic equation.
Aathira: Hi! Bhuvana.
Bhuvana: Hi! Aathira, you look so great!! What's special today?
Aathira: It's my birthday.
Bhuvana: I wish you a happy birthday, Aathira!
Aathira: Thank you, Bhuvana!
Bhuvana: How old are you?
Aathira: I'll give you a hint about my age. Try to find it.
Bhuvana: Ohh! I like solving puzzles. Tell me quickly.
Aathira: I'm two years younger than my brother, and the product of my age and my brother's age is \(80\). Could you find out how old I am?
Bhuvana: Let me take your brother age is \(x\) years. So, your's \(x - 2\) years.
Product of your ages \(=\) \(80\)
\(x (x -2) = 80\)
\(x^2 - 2x = 80\)
\(x^2 - 2x - 80 = 0\)
\(x = -8\) and \(x = 10\)
Bhuvana: Age cannot be a negative number. So, your brother was \(10\) years old, and you are \(10 - 2 = 8\) years old.
Aathira: Aww!!!! That's right!!! How do you find it?
Bhuvana: I converted your hints to a quadratic equation and found your age.
In the above conversation, we have seen a real-life scenario of applying a quadratic equation.
1. Check whether the equation \((x - 3)^2 + 2 = 3x - 4\) is a quadratic or not.
Solution:
\((x - 3)^2 + 2 = 3x - 4\)
\(\Rightarrow x^2 - 6x + 9 + 2 = 3x - 4\)
\(\Rightarrow x^2 - 6x + 11 - 3x + 4 = 0\)
\(\Rightarrow x^2 - 9x + 15 = 0\)
It is of the form \(ax^2 + bx + c = 0\).
Therefore, the given equation is a quadratic equation.
2. Check whether the equation \(x(x + 2) + 6 = x^2 - 3x + 4\) is a quadratic or not.
Solution:
\(x(x + 2) + 6 = x^2 - 3x + 4\)
\(\Rightarrow x^2 + 2x + 6 = x^2 - 3x + 4\)
\(\Rightarrow x^2 + 2x + 6 - x^2 + 3x - 4 = 0\)
\(\Rightarrow 5x + 2 = 0\)
Here, the degree of the equation is \(1\).
Therefore, the given equation is not a quadratic equation.
Reference:
<a href='https://www.freepik.com/photos/birthday'>Birthday photo created by freepik - www.freepik.com</a>