Factorise 2x Squared 7x 3

Factorising Quadratic Expressions: A Deep Dive into 2x² + 7x + 3

Factorising quadratic expressions is a fundamental skill in algebra, crucial for solving equations, simplifying expressions, and understanding many mathematical concepts. This article will provide a comprehensive guide to factorising the specific quadratic expression 2x² + 7x + 3, explaining the process step-by-step and exploring the underlying principles. We’ll also delve into different methods and address frequently asked questions to ensure a thorough understanding. This detailed explanation will allow you to not only solve this particular problem but also confidently tackle other similar quadratic expressions.

Understanding Quadratic Expressions

Before we dive into factorising 2x² + 7x + 3, let's briefly review what quadratic expressions are. 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, and a is not equal to zero. In our case, a = 2, b = 7, and c = 3.

Method 1: The AC Method

The AC method, also known as the splitting the middle term method, is a widely used technique for factorising quadratic expressions. Here's how it works for 2x² + 7x + 3:

1. Find the product AC: Multiply the coefficient of the x² term (a) and the constant term (c). In our example, AC = 2 * 3 = 6.

2. Find two numbers that add up to B and multiply to AC: We need to find two numbers that add up to the coefficient of the x term (b), which is 7, and multiply to 6. These numbers are 6 and 1 (6 + 1 = 7 and 6 * 1 = 6).

3. Rewrite the middle term: Replace the middle term (7x) with the two numbers we found, each multiplied by x. This gives us: 2x² + 6x + 1x + 3.

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

   2x(x + 3) + 1(x + 3)

5. Factor out the common binomial: Notice that both terms now have a common factor of (x + 3). Factor this out:

   (x + 3)(2x + 1)

Therefore, the factorised form of 2x² + 7x + 3 is (x + 3)(2x + 1).

Method 2: Trial and Error

This method involves systematically trying different combinations of binomial factors until you find the one that works. It’s a more intuitive approach but can be time-consuming for more complex quadratics.

Since the coefficient of x² is 2, the factors must be of the form (ax + c)(dx + e), where a and d multiply to 2 and c and e multiply to 3. The possibilities for the factors of 2 are (1, 2) and (-1, -2). The possibilities for factors of 3 are (1, 3) and (-1, -3).

Let’s try some combinations:

• (x + 1)(2x + 3): Expanding this gives 2x² + 5x + 3 (Incorrect)
• (x + 3)(2x + 1): Expanding this gives 2x² + 7x + 3 (Correct!)

Thus, through trial and error, we arrive at the same factorised form: (x + 3)(2x + 1).

Method 3: Quadratic Formula (Indirect Factorisation)

The quadratic formula is a powerful tool for finding the roots (solutions) of a quadratic equation. While not directly a factorisation method, it allows us to indirectly find the factors. The quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

For 2x² + 7x + 3, a = 2, b = 7, and c = 3. Substituting these values into the quadratic formula gives:

x = [-7 ± √(7² - 4 * 2 * 3)] / (2 * 2) x = [-7 ± √(49 - 24)] / 4 x = [-7 ± √25] / 4 x = [-7 ± 5] / 4

This gives two solutions:

x₁ = (-7 + 5) / 4 = -1/2 x₂ = (-7 - 5) / 4 = -3

These solutions represent the roots of the equation 2x² + 7x + 3 = 0. We can use these roots to find the factors. If r and s are the roots of a quadratic equation ax² + bx + c = 0, then the factorised form is a(x - r)(x - s).

Therefore, the factors are: 2(x + 1/2)(x + 3). Simplifying gives us (2x + 1)(x + 3), which is the same as the result obtained using the other methods.

Expanding the Factorised Form (Verification)

To verify our factorisation, we can expand (x + 3)(2x + 1) using the FOIL method (First, Outer, Inner, Last):

• First: x * 2x = 2x²
• Outer: x * 1 = x
• Inner: 3 * 2x = 6x
• Last: 3 * 1 = 3

Combining these terms, we get 2x² + x + 6x + 3 = 2x² + 7x + 3, which confirms our factorisation is correct.

The Significance of Factorisation

Factorising quadratic expressions is more than just a mathematical exercise. It's a crucial skill with wide-ranging applications:

• Solving Quadratic Equations: Factorisation allows us to solve quadratic equations by setting each factor to zero and solving for x. This gives the roots of the equation.
• Simplifying Algebraic Expressions: Factorisation simplifies complex algebraic expressions, making them easier to manipulate and understand.
• Graphing Parabolas: The factors of a quadratic expression reveal the x-intercepts (where the parabola crosses the x-axis) of the corresponding parabolic function.
• Calculus and Beyond: Factorisation forms the foundation for many more advanced mathematical concepts in calculus, differential equations, and other areas.

Frequently Asked Questions (FAQ)

• What if the quadratic expression cannot be factored easily? If the quadratic expression cannot be easily factored using the methods described above, the quadratic formula is a reliable alternative for finding the roots and, indirectly, the factors. Some quadratics may not have real roots, leading to complex factors.

• Are there other methods for factorising quadratics? Yes, there are other methods, such as completing the square, which is particularly useful in cases where the AC method or trial and error prove difficult.

• Why is factorisation important in real-world applications? Factorisation finds applications in various fields, including physics (projectile motion), engineering (designing structures), and economics (modelling growth and decay).

• What if the coefficient of x² is negative? It's often easier to factor out a -1 first before applying any of the methods above. This simplifies the process considerably. For instance, -2x² - 7x - 3 can be rewritten as -1(2x² + 7x + 3), then factored as -1(x+3)(2x+1).

Conclusion

Factorising quadratic expressions is a vital skill in algebra. This article has explored several methods for factorising 2x² + 7x + 3, including the AC method, trial and error, and using the quadratic formula indirectly. Understanding these methods and the underlying principles empowers you to solve similar problems confidently. Remember to practice regularly to build fluency and deepen your understanding of this essential algebraic concept. Mastering factorisation opens doors to more advanced mathematical concepts and their diverse applications in various fields. Don't hesitate to revisit these steps and practice with different quadratic expressions to build your confidence and expertise.