Factorise 3x 2 5x 2

Factorising Quadratic Expressions: A Deep Dive into 3x² + 5x + 2

Factorising quadratic expressions is a fundamental skill in algebra, crucial for solving equations, simplifying expressions, and understanding more advanced mathematical concepts. This comprehensive guide will walk you through the process of factorising the quadratic expression 3x² + 5x + 2, explaining the underlying principles and offering different approaches to tackle similar problems. We'll explore various methods, including the ac method, grouping, and even visual representations, ensuring a thorough understanding regardless of your current mathematical background.

Understanding Quadratic Expressions

Before we delve into factorisation, let's briefly review what a quadratic expression is. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable (usually x) is 2. It takes the general form ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. In our case, the expression 3x² + 5x + 2 fits this form perfectly, with a = 3, b = 5, and c = 2.

The goal of factorising a quadratic expression is to rewrite it as a product of two simpler expressions, typically two linear binomials. This factored form provides valuable insights into the roots (or solutions) of the corresponding quadratic equation (ax² + bx + c = 0).

Method 1: The ac Method (Product-Sum Method)

The ac method, also known as the product-sum method, is a widely used technique for factorising quadratic expressions. It leverages the relationship between the product and sum of the factors of ac.

Steps:

1. Find the product ac: In our example, a = 3 and c = 2, so ac = 3 * 2 = 6.

2. Find two numbers that multiply to ac and add up to b: We need two numbers that multiply to 6 and add up to 5 (our b value). These numbers are 3 and 2 (3 * 2 = 6 and 3 + 2 = 5).

3. Rewrite the middle term: Replace the middle term (5x) with the two numbers found in step 2, expressing them as coefficients of x. This gives us: 3x² + 3x + 2x + 2.

4. Factor by grouping: Group the terms in pairs and factor out the greatest common factor (GCF) from each pair:

   • 3x(x + 1) + 2(x + 1)

5. Factor out the common binomial: Notice that (x + 1) is common to both terms. Factor this out: (x + 1)(3x + 2).

Therefore, the factored form of 3x² + 5x + 2 is (x + 1)(3x + 2).

Method 2: Trial and Error

This method involves a bit of educated guessing but can be quicker once you get the hang of it. It relies on understanding how the binomial factors expand.

Steps:

1. Consider the factors of the leading coefficient (a): The leading coefficient is 3, and its factors are 1 and 3. These will be the coefficients of x in the binomial factors.

2. Consider the factors of the constant term (c): The constant term is 2, and its factors are 1 and 2. These will be the constant terms in the binomial factors.

3. Test different combinations: We need to arrange the factors to achieve the correct middle term (5x). Let's try some possibilities:

   • (x + 1)(3x + 2): Expanding this gives 3x² + 2x + 3x + 2 = 3x² + 5x + 2. This is correct!
   • (x + 2)(3x + 1): Expanding this gives 3x² + x + 6x + 2 = 3x² + 7x + 2. This is incorrect.
   • (3x + 1)(x + 2): Expanding this also gives 3x² + 7x +2

4. Verify: Always expand the factored form to ensure it matches the original quadratic expression.

This method relies on practice and intuition, becoming faster with experience.

Method 3: Completing the Square

While less efficient for this particular example, completing the square is a powerful technique for solving quadratic equations and can also be used to factorise. It's particularly useful when the quadratic doesn't factor easily using the other methods.

This method is more involved and will be explained in detail in a later section to avoid overwhelming the reader at this stage. The focus here is on providing the most efficient methods for this specific problem.

Verification and Checking Your Answer

After factorising, it's crucial to verify your answer. Expand the factored form to ensure it equals the original expression. In our case:

(x + 1)(3x + 2) = 3x² + 2x + 3x + 2 = 3x² + 5x + 2

The expansion matches the original expression, confirming that our factorisation is correct.

Solving Quadratic Equations using Factorised Form

Once you've factorised a quadratic expression, you can easily solve the corresponding quadratic equation. For example, to solve 3x² + 5x + 2 = 0, we use the factored form:

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

This equation is satisfied if either (x + 1) = 0 or (3x + 2) = 0. Solving these linear equations gives us:

• x + 1 = 0 => x = -1
• 3x + 2 = 0 => 3x = -2 => x = -2/3

Therefore, the solutions to the quadratic equation 3x² + 5x + 2 = 0 are x = -1 and x = -2/3. These are also known as the roots of the quadratic equation.

Expanding on the Methods: More Complex Examples and Challenges

The ac method and trial and error are robust techniques applicable to a wider range of quadratic expressions. However, some expressions might present unique challenges:

• Leading coefficient greater than 1: As seen in our example, a leading coefficient greater than 1 introduces more possibilities, requiring careful consideration of factors.
• Negative coefficients: Negative coefficients in a, b, or c require careful attention to signs when applying the ac method or trial and error.
• Expressions that don't factor easily: Some quadratic expressions cannot be factored using integers. In such cases, the quadratic formula or completing the square becomes necessary.

Completing the Square Method (Detailed Explanation)

The completing the square method is a powerful algebraic technique used to solve quadratic equations and can be adapted for factorisation. Let's see how it works:

Steps:

1. Ensure the leading coefficient is 1: If the coefficient of x² (a) is not 1, divide the entire equation by a. In our example, we'd have:

x² + (5/3)x + (2/3) = 0

2. Move the constant term to the right side: Subtract (2/3) from both sides:

x² + (5/3)x = -(2/3)

3. Complete the square: Take half of the coefficient of x ((5/3)/2 = 5/6), square it ((5/6)² = 25/36), and add it to both sides:

x² + (5/3)x + 25/36 = -(2/3) + 25/36

4. Factor the left side as a perfect square: The left side is now a perfect square trinomial:

(x + 5/6)² = 1/36

5. Solve for x: Take the square root of both sides:

x + 5/6 = ±1/6

6. Find the solutions:

• x + 5/6 = 1/6 => x = -4/6 = -2/3
• x + 5/6 = -1/6 => x = -6/6 = -1

This method confirms our previous solutions. To use it for factorisation, reverse the process, working from the perfect square back to the original expression.

Frequently Asked Questions (FAQ)

• Q: What if the quadratic expression doesn't factorise easily?

  *A: If the ac method or trial and error don't yield integer factors, the quadratic formula or completing the square is necessary to find the roots. The expression might not factorise neatly using integers.

• Q: Can I use the quadratic formula to directly find the factors?

  *A: While the quadratic formula gives you the roots, you can use these roots to construct the factors. If the roots are α and β, the factored form is a(x - α)(x - β).

• Q: Is there a visual way to understand factorisation?

  *A: While less common, some visual representations using area models can be helpful, particularly for those who learn best visually. These models illustrate how the terms in the expanded form relate to the dimensions of a rectangle representing the factored form.

Conclusion

Factorising quadratic expressions is a fundamental algebraic skill. This guide has explored multiple methods—the ac method, trial and error, and completing the square—providing a solid understanding of the underlying principles. Remember to always verify your answer by expanding the factored form. Mastering these techniques will unlock your ability to solve quadratic equations and tackle more complex algebraic problems with confidence. The more you practice, the quicker and more intuitively you will be able to factorise these expressions. Remember that perseverance is key, and don't be afraid to experiment with different methods until you find the one that suits your learning style best.