Factorise 3x 2 10x 8

Factorising Quadratic Expressions: A Deep Dive into 3x² + 10x + 8

Factorising quadratic expressions is a fundamental skill in algebra, crucial for solving equations, simplifying expressions, and understanding various mathematical concepts. We'll cover different methods, address common challenges, and answer frequently asked questions. On top of that, this article will provide a practical guide to factorising the quadratic expression 3x² + 10x + 8, explaining the process step-by-step and exploring the underlying mathematical principles. By the end, you'll not only be able to factorise this specific expression but also gain the confidence to tackle other quadratic expressions with ease.

Understanding Quadratic Expressions

Before we get into factorising 3x² + 10x + 8, let's establish a foundational understanding. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable (usually 'x') is 2. It generally takes the form ax² + bx + c, where 'a', 'b', and 'c' are constants. In our case, a = 3, b = 10, and c = 8. Factorising a quadratic expression means rewriting it as a product of two linear expressions (expressions of degree one). This process essentially reverses the expansion of brackets.

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

This is a widely used method for factorising quadratic expressions, particularly effective when the coefficient of x² (the 'a' value) is not equal to 1.

Steps:

1. Find the product 'ac': Multiply the coefficient of x² (a = 3) and the constant term (c = 8). This gives us 3 * 8 = 24.

2. Find two numbers that add up to 'b' and multiply to 'ac': We need two numbers that add up to 10 (the coefficient of x, which is 'b') and multiply to 24. These numbers are 6 and 4 (6 + 4 = 10 and 6 * 4 = 24).

3. Rewrite the middle term: Rewrite the middle term (10x) as the sum of the two numbers found in step 2, using x as the variable. This gives us 6x + 4x.

4. Factor by grouping: Rewrite the original expression using the rewritten middle term: 3x² + 6x + 4x + 8. Now, group the terms in pairs and factor out the greatest common factor (GCF) from each pair:

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

5. Factor out the common binomial: Notice that both terms now share the common factor (x + 2). Factor this out:

   (x + 2)(3x + 4)

Which means, the factorised form of 3x² + 10x + 8 is (x + 2)(3x + 4).

Method 2: Trial and Error

This method is faster when the coefficients are smaller and you can easily spot the factors. It involves systematically trying different combinations of factors until you find the correct one.

1. Consider factors of 'a' and 'c': The coefficient of x² (3) has factors 1 and 3. The constant term (8) has factors 1 and 8, 2 and 4.

2. Test different combinations: We systematically try various combinations of these factors within two binomial brackets, aiming for the correct middle term (10x) when expanding. Let's try some combinations:

   • (x + 1)(3x + 8) → expands to 3x² + 11x + 8 (Incorrect)
   • (x + 8)(3x + 1) → expands to 3x² + 25x + 8 (Incorrect)
   • (x + 2)(3x + 4) → expands to 3x² + 10x + 8 (Correct!)
   • (x + 4)(3x + 2) → expands to 3x² + 14x + 8 (Incorrect)

This method relies on experience and intuition. It may require more time compared to the AC method, especially with larger coefficients.

Method 3: Using the Quadratic Formula (for finding roots)

While not directly factorising, the quadratic formula can help find the roots (solutions) of the quadratic equation 3x² + 10x + 8 = 0. These roots can then be used to construct the factorised form.

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The quadratic formula is: x = [-b ± √(b² - 4ac)] / 2a

For our equation:

• a = 3
• b = 10
• c = 8

Substituting these values into the quadratic formula gives:

x = [-10 ± √(10² - 4 * 3 * 8)] / (2 * 3) x = [-10 ± √(100 - 96)] / 6 x = [-10 ± √4] / 6 x = (-10 ± 2) / 6

This gives two solutions:

x₁ = (-10 + 2) / 6 = -8 / 6 = -4/3 x₂ = (-10 - 2) / 6 = -12 / 6 = -2

The factorised form can then be written as: a(x - x₁)(x - x₂), where 'a' is the coefficient of x² (3 in this case).

Therefore: 3(x + 4/3)(x + 2) = (3x + 4)(x + 2)

Checking Your Answer

Regardless of the method used, always check your answer by expanding the factorised form. Expanding (x + 2)(3x + 4) gives:

x(3x + 4) + 2(3x + 4) = 3x² + 4x + 6x + 8 = 3x² + 10x + 8

This confirms that our factorisation is correct.

Dealing with More Complex Quadratic Expressions

The principles discussed above apply to all quadratic expressions, but the complexity increases with larger coefficients and more challenging numbers. For expressions where finding the appropriate factors through trial and error is difficult, the AC method remains a reliable and reliable approach. Remember to always check your work by expanding the factored form.

The Importance of Factorising

Factorising quadratic expressions is not just a standalone algebraic manipulation. It’s a crucial skill that underpins more advanced mathematical concepts:

• Solving Quadratic Equations: Factorising allows you to solve quadratic equations by setting each factor to zero and solving the resulting linear equations.
• Simplifying Algebraic Fractions: Factorising is essential for simplifying rational expressions (fractions containing polynomials) by cancelling common factors.
• Graphing Quadratic Functions: The factored form of a quadratic helps to identify the x-intercepts (roots) of the parabola represented by the function.
• Calculus: Factorisation plays a role in techniques like finding derivatives and integrals.

Frequently Asked Questions (FAQ)

• Q: What if I can't find the factors easily? A: Use the AC method. It's a systematic approach that works even with more challenging numbers.

• Q: Is there only one correct way to factorise a quadratic? A: While the order of factors might differ (e.g., (x+2)(3x+4) is the same as (3x+4)(x+2)), there is only one unique factored form (ignoring the order of factors).

• Q: What if the quadratic expression cannot be factorised using integers? A: Some quadratic expressions cannot be factorised using integers; these are often solved using the quadratic formula or other numerical methods.

Conclusion

Factorising quadratic expressions is a fundamental skill in algebra with far-reaching applications in mathematics and beyond. In real terms, remember, practice makes perfect! Worth adding: the more you practice factorising, the more intuitive and efficient you will become. Whether you choose the AC method, trial and error, or apply the quadratic formula, always check your answer by expanding to confirm its accuracy. Mastering this skill provides a strong foundation for tackling more complex algebraic problems. So grab some more quadratic expressions and put your newly acquired skills to the test!