Which Is The Factorization Of 8x2 13x 6

Factoring the Quadratic Expression: 8x² + 13x + 6

Factoring quadratic expressions is a fundamental skill in algebra. It's a process that allows us to rewrite a quadratic expression as a product of two simpler expressions (binomials). This skill is crucial for solving quadratic equations, simplifying algebraic fractions, and understanding the behavior of parabolic curves. This article will delve deep into the factorization of the specific quadratic expression 8x² + 13x + 6, exploring multiple methods and providing a comprehensive understanding of the underlying principles.

Understanding Quadratic Expressions

Before we tackle the factorization of 8x² + 13x + 6, let's review the general form of a quadratic expression: ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. In our specific case, a = 8, b = 13, and c = 6.

The goal of factoring is to find two binomials (expressions with two terms) whose product is equal to the original quadratic expression. This can be represented as:

(px + q)(rx + s) = ax² + bx + c

where p, q, r, and s are constants we need to determine.

Method 1: AC Method (Trial and Error)

This method involves finding two numbers that add up to 'b' (13 in our case) and multiply to 'ac' (8 * 6 = 48). Let's list the factor pairs of 48:

• 1 and 48
• 2 and 24
• 3 and 16
• 4 and 12
• 6 and 8

From this list, we see that 3 and 16 add up to 13. Now, we rewrite the middle term (13x) using these two numbers:

8x² + 3x + 16x + 6

Next, we factor by grouping:

Grouping: This technique involves separating the expression into two groups and factoring out the greatest common factor (GCF) from each group.

(8x² + 3x) + (16x + 6)

Now, let's factor out the GCF from each group:

x(8x + 3) + 2(8x + 3)

Notice that both terms now have a common factor of (8x + 3). We can factor this out:

(8x + 3)(x + 2)

Therefore, the factorization of 8x² + 13x + 6 is (8x + 3)(x + 2).

Method 2: Using the Quadratic Formula

While primarily used to solve quadratic equations, the quadratic formula can also be used to find the factors. The quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

Plugging in our values (a = 8, b = 13, c = 6):

x = [-13 ± √(13² - 4 * 8 * 6)] / (2 * 8) x = [-13 ± √(169 - 192)] / 16 x = [-13 ± √(-23)] / 16

Since the discriminant (b² - 4ac = -23) is negative, this quadratic equation has no real roots. However, we can still use the roots (which will be complex numbers in this case) to find the factors. The roots would be:

x₁ = [-13 + i√23] / 16 x₂ = [-13 - i√23] / 16

The factors would then be:

(x - x₁)(x - x₂) = (x - [-13 + i√23]/16)(x - [-13 - i√23]/16)

This method is less efficient for finding the factors of this particular quadratic, because the roots are complex. The AC method is significantly more straightforward for this example.

Method 3: Reverse FOIL (Trial and Error with Binomial Expansion)

This method relies on understanding how binomial multiplication works. Recall the FOIL method (First, Outer, Inner, Last):

(px + q)(rx + s) = prx² + (ps + qr)x + qs

We need to find values for p, q, r, and s that satisfy the equation:

pr = 8 (coefficient of x²) ps + qr = 13 (coefficient of x) qs = 6 (constant term)

We systematically test different factor pairs of 8 and 6:

• Factors of 8: (1, 8), (2, 4)
• Factors of 6: (1, 6), (2, 3), (3,2), (6,1)

Let's try different combinations:

• (x + 2)(8x + 3): This gives 8x² + 3x + 16x + 6 = 8x² + 19x + 6 (Incorrect)
• (2x + 2)(4x + 3): This gives 8x² + 6x + 8x + 6 = 8x² + 14x + 6 (Incorrect)
• (8x + 3)(x + 2): This gives 8x² + 16x + 3x + 6 = 8x² + 19x + 6 (Incorrect)
• (8x+2)(x+3): 8x²+26x+6 (Incorrect)
• (4x+3)(2x+2): 8x²+16x+6 (Incorrect)
• (8x+6)(x+1): 8x²+14x+6 (Incorrect)
• (8x+1)(x+6): 8x²+49x+6 (Incorrect)
• (8x + 3)(x + 2): This gives 8x² + 16x + 3x + 6 = 8x² + 19x + 6 (Incorrect - there was a mistake in the previous attempt. Let's check again)
• (8x + 3)(x + 2): This gives 8x² + 16x + 3x + 6 = 8x² + 19x + 6 (Oops there was an error in my previous calculation - corrected below)
• (8x + 3)(x + 2) = 8x² + 16x + 3x + 6 = 8x² + 19x + 6 (Incorrect).

Let's review the combinations again:

We need to find combinations that add up to 13 and multiply to 48 (8*6). We found that 3 and 16 add up to 19, not 13, meaning that it seems there was an error in the previous calculations. Let's try (x+2)(8x+3):

(x+2)(8x+3) = 8x²+3x+16x+6 = 8x²+19x+6.

The correct combination of factors are:

(8x+3)(x+2) = 8x²+16x+3x+6 = 8x²+19x+6.

There seems to be an error in the above calculation. Let's re-check.

Let's go back to the AC method. We found that 3 and 16 add up to 19, not 13. It seems we made a calculation error before.

We need factors of 48 that add to 13. There are no integer factors that do this.

Let's re-examine the question. It's possible there's a typo in the original expression 8x² + 13x + 6. If the constant term were a different number, factoring would be possible with integer coefficients.

Conclusion

Factoring quadratic expressions can be achieved through various methods, including the AC method, the quadratic formula, and the reverse FOIL method. The AC method and reverse FOIL are generally more efficient for finding integer factors when they exist. The quadratic formula is a powerful tool, but it's less direct for finding factors and can lead to complex roots if the discriminant is negative. The original question might contain a typo in the provided expression 8x²+13x+6, rendering it unfactorable with integers. Double-checking the original expression is essential. Understanding these methods allows for greater flexibility in tackling different types of quadratic expressions and solving related algebraic problems.