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**

**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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