Methods of Solving Quadratic Equations

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

The quadratic equations of the form ax\(^{2}\) + 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 ax\(^{2}\) + bx + c = 0, follow these steps:

Step I: Factorize ax\(^{2}\) + 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

ax\(^{2}\) + bx + c = 0, (where a ≠  0) ………………… (i)

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

4a\(^{2}\)x\(^{2}\) + 4abx + 4ac = 0

⟹ (2ax)\(^{2}\) + 2 . 2ax . b + b\(^{2}\) + 4ac - b\(^{2}\) = 0

⟹ (2ax + b)\(^{2}\) = b\(^{2}\) - 4ac [on simplification and transposition]

Now taking square roots on both sides we get

2ax + b = \(\pm \sqrt{b^{2} - 4ac}\))

⟹ 2ax = -b \(\pm \sqrt{b^{2} - 4ac}\))

⟹ x = \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\)

i.e., \(\frac{-b + \sqrt{b^{2} - 4ac}}{2a}\) or, \(\frac{-b - \sqrt{b^{2} - 4ac}}{2a}\)

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

That means, two roots are obtained for the equation, one is x = \(\frac{-b + \sqrt{b^{2} - 4ac}}{2a}\) and the other is x = \(\frac{-b - \sqrt{b^{2} - 4ac}}{2a}\)


Example to Solving quadratic equation applying factorization method:

Solve the quadratic equation 3x\(^{2}\) - x - 2 = 0 by factorization method.

Solution:

3x\(^{2}\) - x - 2 = 0

Breaking the middle term we get,

⟹ 3x\(^{2}\) - 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 = -\(\frac{2}{3}\)

Therefore, we get x = -\(\frac{2}{3}\), 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 = \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\)

In words, x = \(\frac{-(coefficient  of  x) \pm \sqrt{(coefficient  of  x)^{2} – 4(coefficient  of  x^{2})(constant  term)}}{2  ×  coefficient  of  x^{2}}\)

Proof:

Quadratic equation in general form is

ax\(^{2}\) + bx + c = 0, (where a ≠  0) ………………… (i)

Dividing both sides by a, we get

⟹ x\(^{2}\) + \(\frac{b}{a}\)x + \(\frac{c}{a}\) = 0,

⟹ x\(^{2}\) + 2 \(\frac{b}{2a}\)x + (\(\frac{b}{2a}\))\(^{2}\)  - (\(\frac{b}{2a}\))\(^{2}\)  + \(\frac{c}{a}\) = 0

⟹ (x + \(\frac{b}{2a}\))\(^{2}\) - (\(\frac{b^{2}}{4a^{2}}\) - \(\frac{c}{a}\)) = 0

⟹ (x + \(\frac{b}{2a}\))\(^{2}\) - \(\frac{b^{2} - 4ac}{4a^{2}}\) = 0

⟹ (x + \(\frac{b}{2a}\))\(^{2}\) = \(\frac{b^{2} - 4ac}{4a^{2}}\)

⟹ x + \(\frac{b}{2a}\) = ± \(\sqrt{\frac{b^{2} - 4ac}{4a^{2}}}\)

⟹ x = -\(\frac{b}{2a}\)  ± \(\frac{\sqrt{b^{2} - 4ac}}{2a}\)

⟹ x = \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\)

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 6x\(^{2}\) - 7x + 2 = 0 by applying quadratic formula.

Solution:

6x\(^{2}\) - 7x + 2 = 0

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

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

Now apply Sreedhar Achary’s formula:

x = \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\)

⟹ x = \(\frac{-(-7) \pm \sqrt{(-7)^{2} - 4 ∙ 6 ∙ 2}}{2 × 6}\)

⟹ x = \(\frac{7 \pm \sqrt{49 - 48}}{12}\)

⟹ x = \(\frac{7 \pm 1}{12}\)

Thus, x = \(\frac{7 + 1}{12}\) or, \(\frac{7 - 1}{12}\)

⟹ x = \(\frac{8}{12}\) or, \(\frac{6}{12}\)

⟹ x = \(\frac{2}{3}\) or, \(\frac{1}{2}\)

Therefore, the solutions are x = \(\frac{2}{3}\) or, \(\frac{1}{2}\)

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

From Methods of Solving Quadratic Equations to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Worksheet on Triangle | Homework on Triangle | Different types|Answers

    Jun 21, 24 02:19 AM

    Find the Number of Triangles
    In the worksheet on triangle we will solve 12 different types of questions. 1. Take three non - collinear points L, M, N. Join LM, MN and NL. What figure do you get? Name: (a)The side opposite to ∠L…

    Read More

  2. Worksheet on Circle |Homework on Circle |Questions on Circle |Problems

    Jun 21, 24 01:59 AM

    Circle
    In worksheet on circle we will solve 10 different types of question in circle. 1. The following figure shows a circle with centre O and some line segments drawn in it. Classify the line segments as ra…

    Read More

  3. Circle Math | Parts of a Circle | Terms Related to the Circle | Symbol

    Jun 21, 24 01:30 AM

    Circle using a Compass
    In circle math the terms related to the circle are discussed here. A circle is such a closed curve whose every point is equidistant from a fixed point called its centre. The symbol of circle is O. We…

    Read More

  4. Circle | Interior and Exterior of a Circle | Radius|Problems on Circle

    Jun 21, 24 01:00 AM

    Semi-circular Region
    A circle is the set of all those point in a plane whose distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is known

    Read More

  5. Quadrilateral Worksheet |Different Types of Questions in Quadrilateral

    Jun 19, 24 09:49 AM

    In math practice test on quadrilateral worksheet we will practice different types of questions in quadrilateral. Students can practice the questions of quadrilateral worksheet before the examinations

    Read More