How do you split the middle term in quadratic equations?

The main idea behind splitting the middle term is to rewrite a quadratic equation in a way that makes it easier to factor. Factoring is a key technique for solving quadratic equations. To split the middle term (bx) in ax² + bx + c, find two numbers that multiply to ac and add up to b. Rewrite bx using these numbers, then factor by grouping. For example, in x² + 5x + 6, the numbers 2 and 3 multiply to 6 and add to 5. So, x² + 5x + 6 becomes x² + 2x + 3x + 6, which factors to (x+2)(x+3).

Complete Answer:

Here is the answer to the question How do you split the middle term in quadratic equations?

Splitting the middle term is a method used to factorize quadratic equations of the form a x 2 + b x + c ax^2 + bx + c . This technique involves expressing the middle term b x bx as the sum of two terms whose coefficients multiply to a × c a \times c and add up to b b . Here’s a step-by-step guide:

1. Arrange the Quadratic Expression: Ensure the quadratic expression is in standard form a x 2 + b x + c ax^2 + bx + c .
2. Identify the Product a c ac : Multiply the coefficient of x 2 x^2 (which is a a ) by the constant term c c .
3. Find Two Numbers: Determine two numbers p p and q q such that:
   • p + q = b p + q = b (the coefficient of x x )
   • p × q = a c p \times q = ac
4. Split the Middle Term: Rewrite the middle term b x bx as p x + q x px + qx .
5. Group and Factor: Group the terms into two pairs and factor out the common factors from each pair.
6. Factor the Quadratic Expression: If done correctly, a common binomial factor will emerge, allowing you to write the quadratic expression as a product of two binomials.

Example:

Factorize 6 x 2 + 11 x + 3 6x^2 + 11x + 3 :

1. Standard Form: The expression is already in standard form 6 x 2 + 11 x + 3 6x^2 + 11x + 3 .
2. Calculate a c ac : a = 6 a = 6 , c = 3 c = 3 ; thus, a c = 6 × 3 = 18 ac = 6 \times 3 = 18 .
3. Find p p and q q : We need two numbers that add up to 11 11 and multiply to 18 18 . These numbers are 9 9 and 2 2 because 9 + 2 = 11 9 + 2 = 11 and 9 × 2 = 18 9 \times 2 = 18 .
4. Split the Middle Term: Rewrite 11 x 11x as 9 x + 2 x 9x + 2x : 6 x 2 + 9 x + 2 x + 3 6x^2 + 9x + 2x + 3
5. Group and Factor: Group the terms and factor out the common factors: 3 x ( 2 x + 3 ) + 1 ( 2 x + 3 ) 3x(2x + 3) + 1(2x + 3)
6. Factor the Quadratic Expression: Notice the common factor ( 2 x + 3 ) (2x + 3) : ( 2 x + 3 ) ( 3 x + 1 ) (2x + 3)(3x + 1)

Therefore, 6 x 2 + 11 x + 3 6x^2 + 11x + 3 factors to ( 2 x + 3 ) ( 3 x + 1 ) (2x + 3)(3x + 1) .

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