Quadratic Equations:Part2

 Finding the roots of a quadratic equation using the factorization method

We take a quadratic equation:

x^2 + 5x + 6 = 0

First, we check whether there is any coefficient of  x2.

This means we see if any number other than 1 is multiplied with .

If the coefficient of is 1, then we factor the last number (6) in such a way that the sum of the factors is 5 (or −5, depending on the sign of the middle term).

Finding the Roots of a Quadratic Equation by Factorization Method

We take a quadratic equation:

x^2 + 5x + 6 = 0

First, we check the coefficient of x^2.

If the coefficient of x^2 is 1, we factor the last term in such a way that the sum of the factors is equal to the coefficient of x.

The factors of 6 are 2 and 3, and:

2 + 3 = 5

So, the equation becomes:

x^2 + 3x + 2x + 6 = 0

Now taking common factors:

x(x + 3) + 2(x + 3) = 0

(x + 3)(x + 2) = 0

Equating each factor to zero:

x + 3 = 0

x = -3

x + 2 = 0

x = -2

Hence, the roots of the equation are:

x = -3 and x = -2

When the coefficient of is not 1

Example:

Consider the quadratic equation:

2x^2 + 7x + 3 = 0

Step 1:

Multiply the coefficient of x² with the constant term.

2 × 3 = 6

Step 2:

Find two numbers whose product is 6 and sum is 7.

These numbers are 6 and 1.

Step 3:

Split the middle term using these numbers.

2x^2 + 6x + x + 3 = 0

Step 4:

Take common factors.

2x(x + 3) + 1(x + 3) = 0

Step 5:

Factorize.

(2x + 1)(x + 3) = 0

Step 6:

Equate each factor to zero.

Equate each factor to zero.

2x + 1 = 0

x = -1/2

x + 3 = 0

x = -3

Hence, the roots of the equation are:

x = -1/2 and x = -3

Practice Set: Quadratic Equations (Factorization Method)

Solve the following quadratic equations:

1. x^2 + 7x + 10 = 0

2. x^2 + 9x + 20 = 0

3. x^2 + 11x + 24 = 0

4. 2x^2 + 5x + 3 = 0

5. 3x^2 + 11x + 6 = 0

6. 4x^2 + 12x + 9 = 0

7. 2x^2 − 7x + 3 = 0

Here are only the answers 👍

1. x = −5, −2

2. x = −5, −4

3. x = −3, −8

4. x = −1, −3/2

5. x = −2, −1

6. x = −3/2, −3/2

7. x = 3, 1/2

8. x = 2, 6

By: SYED DANISH SHAHEZAD


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