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Methods•6 min read

How to Solve by Factorization – Splitting the Middle Term Method

By quadraticMaster•

How to Solve by Factorization – Splitting the Middle Term Method

1. Introduction In this lesson, you will learn the factorization method for solving quadratic equations. Also known as "splitting the middle term," this method is a fast and elegant way to find the roots of a quadratic equation when they are integers or simple rational numbers.

2. Core Concept Explanation The factorization method is based on the Zero Product Property, which states that if the product of two quantities is zero, then at least one of the quantities must be zero: \text{If } A \cdot B = 0, \text{ then } A = 0 \text{ or } B = 0

For a quadratic equation in standard form ax^2 + bx + c = 0, we aim to rewrite the expression ax^2 + bx + c as a product of two linear binomials: (dx + e)(fx + g) = 0

Once in this form, we apply the zero product property to find the roots.

3. Step-by-Step Method To solve ax^2 + bx + c = 0 by splitting the middle term: 1. Identify the coefficients a, b, and c. 2. Calculate the product ac. 3. Find two numbers, p and q, that multiply to ac and add to b: p \cdot q = ac \quad \text{and} \quad p + q = b 4. Rewrite the middle term bx as px + qx. 5. Factor by grouping: group the first two terms and the last two terms, factor out the common terms from each group, and then factor out the common binomial. 6. Equate each linear factor to zero and solve for x.

4. Worked Examples * Example 1: Solve x^2 - 5x + 6 = 0. * Solution: 1. Coefficients: a = 1, b = -5, c = 6. Product ac = 6. Sum b = -5. 2. The two numbers are -2 and -3 (since -2 \cdot -3 = 6 and -2 + -3 = -5). 3. Rewrite: x^2 - 2x - 3x + 6 = 0. 4. Group: (x^2 - 2x) - (3x - 6) = 0 \Rightarrow x(x - 2) - 3(x - 2) = 0. 5. Factor: (x - 3)(x - 2) = 0. 6. Roots: x = 3 or x = 2. * Example 2: Solve 2x^2 + 5x - 3 = 0. * Solution: 1. Coefficients: a = 2, b = 5, c = -3. Product ac = -6. Sum b = 5. 2. The two numbers are 6 and -1 (since 6 \cdot -1 = -6 and 6 + -1 = 5). 3. Rewrite: 2x^2 + 6x - x - 3 = 0. 4. Group: 2x(x + 3) - 1(x + 3) = 0. 5. Factor: (2x - 1)(x + 3) = 0. 6. Roots: 2x - 1 = 0 \Rightarrow x = 0.5 or x + 3 = 0 \Rightarrow x = -3. * Example 3: Solve 3x^2 - 10x + 8 = 0. * Solution: 1. Coefficients: a = 3, b = -10, c = 8. Product ac = 24. Sum b = -10. 2. The two numbers are -6 and -4 (since -6 \cdot -4 = 24 and -6 + -4 = -10). 3. Rewrite: 3x^2 - 6x - 4x + 8 = 0. 4. Group: 3x(x - 2) - 4(x - 2) = 0. 5. Factor: (3x - 4)(x - 2) = 0. 6. Roots: 3x - 4 = 0 \Rightarrow x = \frac{4}{3} or x - 2 = 0 \Rightarrow x = 2.

5. Exam Relevance * Example WAEC Question: Solve the equation x^2 - 7x + 10 = 0 by factorization. * Solution: The factors of 10 that add up to -7 are -2 and -5. x^2 - 2x - 5x + 10 = 0 \Rightarrow x(x - 2) - 5(x - 2) = 0 (x - 5)(x - 2) = 0 Roots: x = 5 or x = 2.

6. Common Mistakes * Wrong Sign Combinations: Choosing numbers that multiply to ac but do not add to b. For example, choosing 6 and 1 to factor x^2 - 5x + 6 = 0. While 6 + 1 \neq -5, -2 and -3 is the correct pair. * Incomplete Solutions: Stopping at the factored form (x - p)(x - q) = 0 without writing down the roots x = p or x = q.

7. Practice Questions 1. Solve x^2 - x - 12 = 0 by factorization. * Answer: x = 4 or x = -3. 2. Solve 4x^2 - 12x + 9 = 0. * Answer: x = 1.5 (repeated root).

8. Summary / Cheat Sheet * Zero Product Property: If A \cdot B = 0, then A=0 or B=0. * Splitting Rule: Find p, q where pq = ac and p+q = b. * Remember to solve for x after finding the binomial factors.

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