How To Factor 3x 2 5x 2

How to Factor 3x² + 5x + 2

Factoring quadratic expressions is a fundamental skill in algebra. It's the process of rewriting a quadratic expression (like 3x² + 5x + 2) as a product of simpler expressions. This skill is crucial for solving quadratic equations, simplifying rational expressions, and understanding many other algebraic concepts. This comprehensive guide will walk you through factoring 3x² + 5x + 2, explaining the process step-by-step and providing strategies for tackling similar problems.

Understanding Quadratic Expressions

Before we dive into factoring 3x² + 5x + 2, let's review the standard form of a quadratic expression:

ax² + bx + c

where:

• a, b, and c are constants (numbers).
• x is the variable.
• a ≠ 0 (if a were 0, it wouldn't be a quadratic equation).

In our example, 3x² + 5x + 2, we have:

• a = 3
• b = 5
• c = 2

Method 1: Factoring by Trial and Error

This method involves systematically trying different combinations of factors until you find the correct one. It's a great method for simpler quadratic expressions like ours.

Step 1: Identify the factors of 'a' and 'c'.

• The factors of 'a' (3) are 1 and 3.
• The factors of 'c' (2) are 1 and 2.

Step 2: Test different combinations.

We need to find two binomial factors of the form (px + q)(rx + s) such that:

• p * r = a (3 in this case)
• q * s = c (2 in this case)
• p * s + q * r = b (5 in this case)

Let's try some combinations:

• (x + 1)(3x + 2): Expanding this gives 3x² + 2x + 3x + 2 = 3x² + 5x + 2. This works!

• (x + 2)(3x + 1): Expanding this gives 3x² + x + 6x + 2 = 3x² + 7x + 2. This doesn't work.

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

Method 2: AC Method (or Decomposition Method)

The AC method is a more systematic approach that works well for more complex quadratic expressions.

Step 1: Find the product 'ac'.

In our case, ac = 3 * 2 = 6.

Step 2: Find two numbers that add up to 'b' and multiply to 'ac'.

We need two numbers that add up to 5 (our 'b' value) and multiply to 6 (our 'ac' value). These numbers are 3 and 2.

Step 3: Rewrite the middle term ('bx') using these two numbers.

We rewrite 5x as 3x + 2x:

3x² + 3x + 2x + 2

Step 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)

Step 5: Factor out the common binomial factor.

Notice that both terms now share the common factor (x + 1). Factor this out:

(x + 1)(3x + 2)

Again, we arrive at the factored form: (x + 1)(3x + 2).

Method 3: Quadratic Formula (for finding the roots)

While the quadratic formula doesn't directly give you the factored form, it helps you find the roots (the values of x that make the expression equal to zero). These roots can then be used to construct the factored form.

The quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

For 3x² + 5x + 2:

x = [-5 ± √(5² - 4 * 3 * 2)] / (2 * 3) x = [-5 ± √(25 - 24)] / 6 x = [-5 ± √1] / 6 x = (-5 ± 1) / 6

This gives us two solutions:

x = (-5 + 1) / 6 = -4/6 = -2/3 x = (-5 - 1) / 6 = -6/6 = -1

These roots correspond to the factors (3x + 2) and (x + 1). Note that the roots are the values of x that make each factor equal to zero. To obtain the factors from the roots, remember that if x = -a is a root, then (x + a) is a factor.

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

Checking Your Answer

Always check your factored form by expanding it to ensure it matches the original quadratic expression:

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

This confirms that our factoring is correct.

Factoring More Complex Quadratics

The methods outlined above can be applied to more complex quadratic expressions, but the trial and error method might become less efficient. The AC method remains a reliable strategy. Remember to always look for common factors before attempting to factor the quadratic.

Applications of Factoring

Factoring quadratic expressions is essential for various algebraic applications, including:

• Solving quadratic equations: Setting the factored expression equal to zero allows you to easily find the solutions (roots) of the quadratic equation.

• Simplifying rational expressions: Factoring the numerator and denominator can help cancel out common factors, simplifying the expression.

• Graphing quadratic functions: The factored form helps identify the x-intercepts (where the graph crosses the x-axis) of the parabola representing the quadratic function.

• Calculus: Factoring is crucial in calculus for tasks such as finding derivatives and integrals.

Practice Problems

To solidify your understanding, try factoring the following quadratic expressions:

1. 2x² + 7x + 3
2. 4x² - 4x - 3
3. 6x² + 11x - 10
4. x² - 5x + 6

By consistently practicing these methods and working through various examples, you'll develop a strong grasp of factoring quadratic expressions, a key skill in algebra and beyond. Remember that consistent practice is the key to mastering this important algebraic skill. Don't be afraid to try different methods until you find the one that works best for you. Good luck!