General Properties of Quadratic Equation

We will discuss here about some of the general properties of quadratic equation.

We know that the general form of quadratic equation is ax^2 + bx + c = 0, where a is the co-efficient of x^2, b is the coefficient of x, c is the constant term and a ≠ 0, since, if a = 0, then the equation will no longer remain a quadratic

When we are expressing any quadratic equation in the form of ax^2 + bx + c =0, we have in the left side of the equation a quadratic expression.

For example, we can write the quadratic equation x^2 + 3x = 10 as x^2 + 3x – 10 = 0.

Now we will learn how to factorize the above quadratic expression.

x^2 + 3x - 10

= x^2 + 5x  - 2x - 10

= x(x + 5) -2 (x + 5)

= (x + 5)(x – 2),

Therefore, x^2 + 3x – 10 = (x + 5)(x – 2) ............ (A)

Note: We know that mn = 0 implies that, either (i) m = 0 or n = 0 or (ii) m = 0 and n = 0. It is not possible that both of m and n are non-zero.

From (A) we get,

(x + 5)(x – 2) = 0, then any one of x + 5 and x - 2 must be zero.

So, factorizing the left side of the equation x^2 + 3x – 10 = 0 we get, (x + 5)(x – 2) = 0

Therefore, any one of (x + 5) and (x – 2) must be zero

i.e., x + 5 = 0 ................ (I)

or, x – 2 = 0 .................. (II)

Both of (I) and (II) represent linear equations, which we can solve to get the value of x.

From equation (I), we get x = -5 and from equation (II), we get x = 2.

Therefore the solutions of the equation are x = -5 and x = 2.

We will solve a quadratic equation in the following way:

(i) First we need to express the given equation in the general form of the quadratic equation ax^2 + bx + c = 0, then

(ii) We need to factorize the left side of the quadratic equation,

(iii) Now express each of the two factor equals to 0 and solve them

(iv)The two solutions are called the roots of the given quadratic equation.

Notes: (i) If b ≠ 0 and c = 0, one root of the quadratic equation is always zero.

For example, in the equation 2x^2 - 7x = 0, there is no constant term. Now factoring the left side of the equation, we get x(2x - 7).

Therefore, x(2x - 7) = 0.

Thus, either x = 0 or, 2x – 7 = 0

either x = 0 or, x = 7/2

Therefore, the two roots of the equation 2x^2 - 7x = 0 are 0, 7/2.

(ii) If b = 0, c = 0, both the roots of the quadratic equation will be zero. For example, if 11x^2 = 0, then dividing both sides by 11, we get x^2 = 0 or x = 0, 0.

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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