What is Middle Term Splitting?

Middle term splitting is a method used in algebra to factorize quadratic expressions in the form of ax² + bx + c, where a, b, and c are coefficients. This method involves splitting the middle term of the quadratic expression into two terms such that their sum is equal to the coefficient of the middle term and their product is equal to the product of the coefficient of the square term and the constant term.

Step-by-Step Explanation:

To factorize a quadratic expression using middle term splitting, follow these steps:

Step 1: Identify the coefficients
- Identify the coefficients a, b, and c in the quadratic expression ax² + bx + c.

Step 2: Find the product of a and c
- Multiply the coefficient of the square term (a) with the constant term (c) to find the product.

Step 3: Split the middle term
- Look for two numbers that multiply to give the product obtained in the previous step and add up to the coefficient of the middle term (b).
- Let's say the two numbers are p and q.

Step 4: Rewrite the quadratic expression
- Rewrite the quadratic expression by splitting the middle term using the values of p and q.
- Instead of the original middle term (bx), write (px + qx).

Step 5: Group the terms
- Group the terms in pairs.
- (ax² + px) + (qx + c)

Step 6: Factor by grouping
- Factor out the greatest common factor from each group.
- x(a + p) + q(a + p)

Step 7: Factor out the common binomial factor
- Factor out the common binomial factor from both groups.
- (a + p)(x + q)

Step 8: Simplify the expression
- Simplify the expression by multiplying the factors obtained in the previous step.

Example:

Let's consider the quadratic expression 2x² + 7x + 3 and factorize it using middle term splitting.

Step 1: Identify the coefficients
- a = 2, b = 7, c = 3

Step 2: Find the product of a and c
- Product = 2 * 3 = 6

Step 3: Split the middle term
- Find two numbers that multiply to 6 and add up to 7.
- The numbers are 6 and 1.

Step 4: Rewrite the quadratic expression
- 2x² + 6x + x + 3

Step 5: Group the terms
- (2x² + 6x) + (x + 3)

Step 6: Factor by grouping
- 2x(x + 3) + 1(x + 3)

Step 7: Factor out the common binomial factor
- (x + 3)(2x + 1)

Step 8: Simplify the expression
- (x + 3)(2x + 1)

Conclusion:

Middle term

By the help of pen and notebook