Introduction to Quadratic Equation

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.

Quadratic Equation

Introduction to Quadratic Equation

Formation of Quadratic Equation in One Variable

Solving Quadratic Equations

General Properties of Quadratic Equation

Methods of Solving Quadratic Equations

Roots of a Quadratic Equation

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








9th Grade Math

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