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Methods of Solving Quadratic Equations

We will discuss here about the methods of solving quadratic equations.

The quadratic equations of the form ax2 + bx + c = 0 is solved by any one of the following two methods (a) by factorization and (b) by formula.

(a) By factorization method:

In order to solve the quadratic equation ax2 + bx + c = 0, follow these steps:

Step I: Factorize ax2 + bx + c in linear factors by breaking the middle term or by completing square.

Step II: Equate each factor to zero to get two linear equations (using zero-product rule).

Step III: Solve the two linear equations. This gives two roots (solutions) of the quadratic equation.



Quadratic equation in general form is

ax2 + bx + c = 0, (where a ≠  0) ………………… (i)

Multiplying both sides of, ( i) by 4a,

4a2x2 + 4abx + 4ac = 0

⟹ (2ax)2 + 2 . 2ax . b + b2 + 4ac - b2 = 0

⟹ (2ax + b)2 = b2 - 4ac [on simplification and transposition]

Now taking square roots on both sides we get

2ax + b = ±b24ac)

⟹ 2ax = -b ±b24ac)

⟹ x = b±b24ac2a

i.e., b+b24ac2a or, bb24ac2a

Solving the quadratic equation (i), we have got two values of x.

That means, two roots are obtained for the equation, one is x = b+b24ac2a and the other is x = bb24ac2a


Example to Solving quadratic equation applying factorization method:

Solve the quadratic equation 3x2 - x - 2 = 0 by factorization method.

Solution:

3x2 - x - 2 = 0

Breaking the middle term we get,

⟹ 3x2 - 3x + 2x - 2 = 0

⟹ 3x(x - 1) + 2(x - 1) = 0

⟹ (x - 1)(3x + 2) = 0

Now, using zero-product rule we get,

x - 1 = 0 or, 3x + 2 = 0

⟹ x = 1 or x = -23

Therefore, we get x = -23, 1.

These are the two solutions of the equation.

 


(b) By using formula:

To form the Sreedhar Acharya’s formula and use it in solving quadratic equations

The solution of the quadratic equation ax^2 + bx + c = 0 are x = b±b24ac2a

In words, x = (coefficientofx)±(coefficientofx)24(coefficientofx2)(constantterm)2×coefficientofx2

Proof:

Quadratic equation in general form is

ax2 + bx + c = 0, (where a ≠  0) ………………… (i)

Dividing both sides by a, we get

⟹ x2 + bax + ca = 0,

⟹ x2 + 2 b2ax + (b2a)2  - (b2a)2  + ca = 0

⟹ (x + b2a)2 - (b24a2 - ca) = 0

⟹ (x + b2a)2 - b24ac4a2 = 0

⟹ (x + b2a)2 = b24ac4a2

⟹ x + b2a = ± b24ac4a2

⟹ x = -b2a  ± b24ac2a

⟹ x = b±b24ac2a

This is the general formula for finding two roots of any quadratic equation. This formula is known as quadratic formula or Sreedhar Acharya’s formula.

 

Example to Solving quadratic equation applying Sreedhar Achary’s formula:

Solve the quadratic equation 6x2 - 7x + 2 = 0 by applying quadratic formula.

Solution:

6x2 - 7x + 2 = 0

First we need to compare the given equation 6x2 - 7x + 2 = 0 with the general form of the quadratic equation ax2 + bx + c = 0, (where a ≠  0) we get,

a = 6, b = -7 and c =2

Now apply Sreedhar Achary’s formula:

x = b±b24ac2a

⟹ x = (7)±(7)24622×6

⟹ x = 7±494812

⟹ x = 7±112

Thus, x = 7+112 or, 7112

⟹ x = 812 or, 612

⟹ x = 23 or, 12

Therefore, the solutions are x = 23 or, 12

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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