Quadratic Factorization using Splitting of Middle Term

Find the factors of 6x2 - 13x + 6

Steps to find out Quadratic equation are as follows:

1.   Divide 1st factor by 1st term

so the 1st factor will be (3x-2)

1.   Divide 2nd factor  by 1st term

so the 2nd factor will be  (3x-2)

So the factors of quadratic equation will be

6x2 -13x +6 are (2x-3) (3x-2)

Roots of the equation are
3x – 2 = 0 ⇒ 3x = 2 so x = 2/3
2x – 3 = 0 ⇒ 2x = 3 so x = 3/2

Roots are {2/3, 3/2}

12 x² +11x -15 = 0

Solution:

12x2 -15 = - 11x
12x2 -15 + 11x = 0  ( add + 11x )
12x2 + 11x -15 = 0

Steps to find the factors of Quadratic equation are as follows:

1.   Divide the 1st factor by 1st term

so the factor will be (3x + 5)

Now divide the 2nd factor by 1st term

so the factor will be  (4x – 3)

So the factors of quadratic equation

12x2 +11x -15 are (3x + 5) (4x – 3)

Quadratic expressions are expressions with a variable of degree 2, i.e., expressions of the form ax^2 + bx + c, where a, b, and c are constants. The quadratic equation method involves rewriting the quadratic expression as a product of two linear expressions. Here, we will focus on the technique of splitting the middle term, which is used when the leading coefficient 'a' is not equal to 1. This method involves splitting the linear term 'bx' into two terms whose sum is equal to 'bx' and whose product is equal to 'a*c'. We can then factor out the common factors in the resulting expression and write the expression as a product of two linear terms.Let's take an example to understand this method in detail.

Example 1:

Factorize the quadratic expression 6x^2 + 11x + 4 Solution:

Step 1: Multiply the leading coefficient and the constant term, which gives 6 * 4 = 24

Step 2: Find two numbers whose sum is equal to the coefficient of x, which is 11, and whose product is equal to 24. The numbers are 3 and 8, since 3 + 8 = 11, and 3 * 8 = 24

Step 3: Rewrite the quadratic expression by splitting the linear term into two terms using the numbers found in step 2. We can write

6x^2 + 3x + 8x + 4

Step 4: Factor out the common factor in the first two terms, which is 3x, and the common factor in the last two terms, which is 4. This gives us 3x(2x + 1) + 4(2x + 1)

Step 5: Notice that the two terms inside the parentheses are the same. Factor out this common factor to obtain the final factorization

(3x + 4)(2x + 1)

Hence, 6x^2 + 11x + 4 factors into (3x + 4)(2x + 1) Example 2:

Factorize the quadratic expression 2x^2 - 5x - 3.   Solution:

Step 1: Multiply the leading coefficient and the constant term, which gives 2 * (-3) = -6

Step 2: Find two numbers whose sum is equal to the coefficient of x, which is -5, and whose product is equal to -6. The numbers are -3 and 2, since -3 + 2 = -1, and -3 * 2 = -6

Step 3: Rewrite the quadratic expression by splitting the linear term into two terms using the numbers found in step 2. We can write

2x^2 - 3x - 2x - 3

Step 4: Factor out the common factor in the first two terms, which is x, and the common factor in the last two terms, which is -1. This gives us x(2x - 3) - 1(2x - 3)

Step 5: Notice that the two terms inside the parentheses are the same. Factor out this common factor to obtain the final factorization

(x - 1)(2x - 3)

Hence, 2x^2 - 5x - 3 factors into (x - 1)(2x - 3)

In summary, the technique of splitting the middle term involves splitting the linear term into two terms whose sum is equal to the coefficient.

Example 1:

Factorize x^2 + 5x + 6

Solution: To factorize x^2 + 5x + 6, we need to find two numbers whose sum is 5 and product is 6. These numbers are 2 and 3. Therefore, we can split the middle term 5x as 2x + 3x. We can then factor the expression as follows:

x^2 + 5x + 6 = x^2 + 2x + 3x + 6 = (x + 2)(x + 3)

Therefore, x^2 + 5x + 6 can be factorized as (x + 2)(x + 3).

Example 2:

Factorize 2x^2 + 7x + 3

Solution: To factorize 2x^2 + 7x + 3, we need to find two numbers whose sum is 7/2 and product is 3/2. These numbers are 3/2 and 1/2. Therefore, we can split the middle term 7x as 3x/2 + x/2. We can then factor the expression as follows:

2x^2 + 7x + 3 = 2x^2 + 3x/2 + x/2 + 3 = (2x + 1)(x + 3)

Therefore, 2x^2 + 7x + 3 can be factorized as (2x + 1)(x + 3)

Example 3:

Factorize 4x^2 - 4x - 3

Solution: To factorize 4x^2 - 4x - 3, we need to find two numbers whose sum is -4/4 = -1 and product is -3/4. These numbers are -3/2 and 1/2. Therefore, we can split the middle term -4x as -3x - x. We can then factor the equation as follows:

4x^2 - 4x - 3 = 4x^2 - 3x - x - 3 = (4x^2 - 3x) - (x + 3) = x(4x - 3) - (x + 3) = (4x - 3)(x - 1)

Therefore, 4x^2 - 4x - 3 can be factorized as (4x - 3)(x - 1).

These examples illustrate how the technique of splitting the middle term can be used to factorize quadratic expressions into simpler forms. It is a powerful tool that can simplify complex expressions and make them easier to work with.

Solving quadratic equations: One of the most common practical uses of quadratic factorization using splitting of middle term is in solving quadratic equations. By factoring the quadratic expression into two linear expressions, we can easily find the roots of the quadratic equation.

Simplifying algebraic expressions: The technique of quadratic factorization using splitting of middle term can be used to simplify complex algebraic expressions. By factoring out the common factors, we can simplify the expressions and make them easier to work with.

Solving optimization problems: In optimization problems, we often need to find the maximum or minimum value of a quadratic expression. By factoring the give expression together, we can easily identify the critical points and determine the optimal value.

Solving projectile motion problems: In physics, the technique of factoring quadratic equations by factorization using splitting of middle term is used to solve projectile motion problems. By factoring the quadratic equation, we can determine the time of flight, maximum height, and range of a projectile.

Developing computer algorithms: Quadratic factorization using splitting of middle term is used in computer algorithms, such as cryptography and error-correcting codes. By factoring large numbers, we can develop secure encryption algorithms and error-correcting codes.

Overall, the technique of quadratic factorization using splitting of middle term is a powerful tool with practical applications in various fields. It enables us to simplify complex expressions, solve quadratic equations, and solve optimization problems, making it an essential skill for students and professionals alike.