Factorise 3x 2 10x 3

Factorising 3x² + 10x + 3: A thorough look

Factorising quadratic expressions is a fundamental skill in algebra. Worth adding: understanding how to factorise allows you to solve quadratic equations, simplify algebraic expressions, and delve deeper into more complex mathematical concepts. Also, this article provides a thorough explanation of how to factorise the quadratic expression 3x² + 10x + 3, covering various methods and offering insights into the underlying principles. We will explore different approaches, including the traditional method, the AC method, and even consider the use of the quadratic formula in understanding the factors.

Understanding Quadratic Expressions

Before we dive into the factorisation of 3x² + 10x + 3, let's establish a basic understanding of quadratic expressions. Here's the thing — a quadratic expression is a polynomial of degree two, meaning the highest power of the variable (in this case, x) is 2. The general form of a quadratic expression is ax² + bx + c, where a, b, and c are constants. In our example, 3x² + 10x + 3, a = 3, b = 10, and c = 3. Factorising this expression means rewriting it as a product of two simpler expressions, typically two linear binomials.

Method 1: The Traditional Method (Trial and Error)

This method involves finding two binomials (expressions with two terms) that, when multiplied together, give the original quadratic expression. It relies on understanding how to expand binomials using the FOIL (First, Outer, Inner, Last) method.

We're looking for two binomials in the form (px + q)(rx + s) such that:

(px)(rx) = 3x²: This means the product of the 'x' terms must be 3x².
(px)(s) + (q)(rx) = 10x: The sum of the outer and inner products must equal 10x.
(q)(s) = 3: The product of the constant terms must be 3.

Let's consider the possible factors of 3x² and 3:

Factors of 3x²: (3x, x)
Factors of 3: (3, 1) and (1, 3)

Now, let's try different combinations:

(3x + 1)(x + 3): Expanding this gives 3x² + 9x + x + 3 = 3x² + 10x + 3. This works!

That's why, the factorised form of 3x² + 10x + 3 is (3x + 1)(x + 3).

Method 2: The AC Method

The AC method provides a more systematic approach to factorising quadratic expressions, especially those with larger coefficients. It's particularly useful when the trial-and-error method becomes cumbersome.

Find the product AC: In our case, a = 3 and c = 3, so AC = 3 * 3 = 9.
Find two numbers that add up to B and multiply to AC: We need two numbers that add up to 10 (the value of b) and multiply to 9. These numbers are 9 and 1 (9 + 1 = 10 and 9 * 1 = 9).
Rewrite the middle term: Rewrite the original expression using the two numbers found in step 2: 3x² + 9x + x + 3.
Factor by grouping: Group the terms in pairs and factor out the greatest common factor (GCF) from each pair:

3x(x + 3) + 1(x + 3)
Factor out the common binomial: Notice that (x + 3) is a common factor in both terms. Factor it out:

(x + 3)(3x + 1)

The factorised form, as before, is (3x + 1)(x + 3). The order of the factors doesn't matter; (x + 3)(3x + 1) is equivalent to (3x + 1)(x + 3).

Method 3: Using the Quadratic Formula (for understanding factors)

While not a direct factorisation method, the quadratic formula can help us understand the roots of the quadratic equation 3x² + 10x + 3 = 0, which are directly related to the factors. The quadratic formula is:

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x = [-b ± √(b² - 4ac)] / 2a

Substituting the values from our expression:

x = [-10 ± √(10² - 4 * 3 * 3)] / (2 * 3) x = [-10 ± √(100 - 36)] / 6 x = [-10 ± √64] / 6 x = [-10 ± 8] / 6

This gives us two solutions:

x = (-10 + 8) / 6 = -2/6 = -1/3
x = (-10 - 8) / 6 = -18/6 = -3

These solutions represent the roots of the quadratic equation. If r and s are the roots, then the factorised form is a(x - r)(x - s). In our case:

r = -1/3 and s = -3
a = 3

So, the factorised form is:

3(x + 1/3)(x + 3) = (3x + 1)(x + 3)

Why Factorisation Matters

Factorising quadratic expressions is crucial for several reasons:

Solving Quadratic Equations: Setting a quadratic expression equal to zero creates a quadratic equation. Factorisation allows you to find the roots (solutions) of the equation easily by setting each factor equal to zero and solving for x.
Simplifying Expressions: Factorisation simplifies complex algebraic expressions, making them easier to manipulate and understand.
Graphing Quadratic Functions: The factors of a quadratic expression help determine the x-intercepts (where the graph crosses the x-axis) of the corresponding quadratic function.
Further Algebraic Manipulations: Factorisation is a foundation for more advanced algebraic techniques, such as partial fraction decomposition and the solution of higher-degree polynomial equations.

Frequently Asked Questions (FAQ)

Q: What if I can't find the factors easily? A: If the trial-and-error method proves difficult, the AC method provides a more structured approach. Alternatively, you can always use the quadratic formula to find the roots and then construct the factors from the roots.
Q: Is there only one way to factorise a quadratic expression? A: No, there isn't. While there might be several possible factorisations depending on the coefficients, they will all be equivalent when expanded. To give you an idea, –(3x+1)(x+3) is a valid factorisation although it is expressed with a minus sign.
Q: What if the quadratic expression cannot be factorised? A: Some quadratic expressions cannot be factorised using integer coefficients. In such cases, the quadratic formula will provide the roots, but the factors might involve irrational numbers or complex numbers.

Conclusion

Factorising quadratic expressions like 3x² + 10x + 3 is a vital skill in algebra. This article explored multiple methods – the traditional trial-and-error approach, the more systematic AC method, and the insightful application of the quadratic formula. Mastering these techniques will significantly enhance your algebraic abilities and allow you to tackle more complex mathematical problems confidently. That's why remember, practice is key! The more you practice factorising quadratic expressions, the quicker and more proficient you will become. Don't be afraid to try different methods and find the one that works best for your learning style. The understanding of factorisation extends beyond simple algebraic manipulation – it's a fundamental concept with far-reaching implications in various branches of mathematics and its applications.