Factorise 2x 2 7x 15

Factorising Quadratic Expressions: A Deep Dive into 2x² + 7x + 15

Factorising quadratic expressions is a fundamental skill in algebra, crucial for solving equations, simplifying expressions, and understanding various mathematical concepts. This article provides a comprehensive guide to factorising the quadratic expression 2x² + 7x + 15, explaining the process step-by-step, exploring different methods, and addressing common questions. Understanding this process will equip you with the tools to tackle similar problems with confidence.

Understanding Quadratic Expressions

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. Factorising a quadratic expression involves rewriting it as a product of two linear expressions (expressions of degree one). For example, factorising x² + 5x + 6 results in (x + 2)(x + 3). This means that (x+2) multiplied by (x+3) equals x² + 5x + 6.

Method 1: The AC Method for Factorising 2x² + 7x + 15

The AC method is a systematic approach for factorising quadratic expressions of the form ax² + bx + c. It's particularly useful when 'a' is not equal to 1, as in our case (a = 2, b = 7, c = 15).

Steps:

Find the product AC: Multiply the coefficient of x² (a) by the constant term (c): 2 * 15 = 30
Find two numbers that add up to B and multiply to AC: We need two numbers that add up to 7 (the coefficient of x, which is b) and multiply to 30. These numbers are 3 and 10. (3 + 10 = 13, not 7. Let's try another combination) Actually, there is no such combination in this case. This means the quadratic cannot be factorised using integers.
Rewrite the middle term: Since we cannot find such numbers that add up to 7 and multiply to 30, we can try the following method or confirm that this quadratic doesn't factor using integers.

Method 2: Trial and Error Method for 2x² + 7x + 15

This method involves systematically trying different combinations of factors until you find the correct pair. While less structured than the AC method, it can be faster for some quadratics.

Since the coefficient of x² is 2, the linear factors must be of the form (2x + p)(x + q), where p and q are constants. We need to find p and q such that:

(2x + p)(x + q) = 2x² + 7x + 15

Expanding this expression, we get:

2x² + (2q + p)x + pq = 2x² + 7x + 15

Comparing coefficients, we have:

2q + p = 7
pq = 15

Now, let's try different factor pairs of 15:

1 and 15: If p = 1 and q = 15, then 2q + p = 2(15) + 1 = 31 (Incorrect)
3 and 5: If p = 3 and q = 5, then 2q + p = 2(5) + 3 = 13 (Incorrect)
5 and 3: If p = 5 and q = 3, then 2q + p = 2(3) + 5 = 11 (Incorrect)
15 and 1: If p = 15 and q = 1, then 2q + p = 2(1) + 15 = 17 (Incorrect)

It appears that there are no integer values of p and q that satisfy both equations.

Method 3: Using the Quadratic Formula

When factorisation using integers is not possible, we can use the quadratic formula to find the roots of the equation 2x² + 7x + 15 = 0. The quadratic formula is:

x = (-b ± √(b² - 4ac)) / 2a

Substituting a = 2, b = 7, and c = 15:

x = (-7 ± √(7² - 4 * 2 * 15)) / (2 * 2) x = (-7 ± √(49 - 120)) / 4 x = (-7 ± √(-71)) / 4

Since the discriminant (b² - 4ac = -71) is negative, the roots are complex numbers. This confirms that the quadratic expression 2x² + 7x + 15 cannot be factorised into real linear factors using integers.

Why Factorisation is Important

Factorising quadratic expressions is a crucial skill because:

Solving Quadratic Equations: If a quadratic equation is in factorised form, it's easy to find its roots (solutions) by setting each factor to zero.
Simplifying Expressions: Factorisation allows us to simplify complex algebraic expressions, making them easier to work with.
Graphing Quadratic Functions: The factored form of a quadratic helps in determining the x-intercepts (where the graph crosses the x-axis) of the corresponding quadratic function.
Further Algebraic Manipulations: Factorisation is a foundational step for more advanced algebraic techniques like partial fraction decomposition and solving systems of equations.

Frequently Asked Questions (FAQ)

Q: What if the quadratic expression is not factorisable using integers?

A: If the discriminant (b² - 4ac) is negative, the quadratic has no real roots, and it cannot be factorised using real numbers. The roots will be complex numbers. If the discriminant is positive, but the quadratic cannot be factorised using integers, it may still have real roots, which can be found using the quadratic formula.

Q: Are there other methods for factorising quadratic expressions?

A: Yes, there are other approaches, such as completing the square, but the AC method and trial-and-error are generally preferred for their efficiency.

Q: What if I make a mistake during the factorisation process?

A: Always check your answer by expanding the factorised expression to see if it matches the original quadratic. If it doesn't, review your steps carefully.

Conclusion

Factorising the quadratic expression 2x² + 7x + 15 presents a unique case. While the methods described (AC method and trial and error) are powerful tools for factorising many quadratic expressions, they fail to yield integer factors for this specific example. This is because the discriminant is negative, indicating that the quadratic has no real roots and, therefore, cannot be factored using real numbers. Understanding this limitation is as crucial as mastering the techniques themselves. The quadratic formula serves as a valuable tool for finding the roots in such instances, further solidifying your understanding of quadratic equations and their properties. Remember to practice regularly to build your proficiency and confidence in factorising quadratic expressions. The more you practice, the better you'll become at recognizing patterns and choosing the most efficient method for each problem.