We will discuss about the introduction to quadratic equation in details.
Let us start with the following problem:
Suppose, in a school students of class IX collect $ 10.50. Each of them contributing the number of cents, which is 5 more than the number of students in the class.
To express the above statement in mathematical language,
Let the number of students in class IX be x
Each students contributes (x + 5) Cents
Total amount collected from the student = x (x + 5) Cents
According to the problem, total collection is $ 10.50 or 1050 Cents
Now from the given question we get,
x(x + 5) = 1050
⟹ x\(^{2}\) + 5x = 1050
⟹ x\(^{2}\) + 5x  1050 = 0
Therefore, the equation x\(^{2}\) + 5x  1050 = 0 represents the above statement.
The equation x\(^{2}\) + 5x  1050 = 0 is formed of only one variable (unknown quantity) x.
Here, the highest power of x is 2 (two).
This type of equation is called Quadratic Equation.
Definition of Quadratic Equation:
If the highest power of the variable of an equation in one variable is 2, then that equation is called a Quadratic Equation.
Some of the examples of quadratic equations:—
(i) x\(^{2}\)  7x + 12 = 0
(ii) 3x\(^{2}\) – 4x – 4 = 0
(iii) x\(^{2}\) = 16
(iv) (x + 3)(x  3) + 5 = 0
(v) 3z  \(\frac{8}{z}\) = 2
To know the highest power of the variable in an equation, it becomes, sometimes, necessary to simplify the expression involved in the equation.
For example, the highest power of x in the equation \(\frac{x}{4}\) + \(\frac{7}{x}\) = \(\frac{3}{5}\) may appear to be one, but on simplification we get 5x\(^{2}\)  12x + 140 = 0.
So, it is a quadratic equation
Again, 4(3x\(^{2}\)  7x + 5) = 2(4x\(^{2}\)  7x + 4) looks like a quadratic equation, but, it is really a linear equation.
Assuming, x\(^{2}\) = z the equation x\(^{4}\)  3x\(^{2}\) + 7 = 0 reduces to z\(^{2}\)  3z + 7 = 0, which is a quadratic equation.
Hence, the equations involving higher powers can be reduced to a quadratic equation by substitution.
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