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.
Algebra/Linear Algebra
Introduction to Quadratic Equation
Formation of Quadratic Equation in One Variable
General Properties of Quadratic Equation
Methods of Solving Quadratic Equations
Examine the Roots of a Quadratic Equation
Problems on Quadratic Equations
Quadratic Equations by Factoring
Word Problems Using Quadratic Formula
Examples on Quadratic Equations
Word Problems on Quadratic Equations by Factoring
Worksheet on Formation of Quadratic Equation in One Variable
Worksheet on Quadratic Formula
Worksheet on Nature of the Roots of a Quadratic Equation
Worksheet on Word Problems on Quadratic Equations by Factoring
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