Problem 46 - Algebra and Trigonometry

Question

Factor the expression completely. $$ 8 x^{2}+10 x+3 $$

Step-by-Step Solution

Answer

The expression factors as \((4x + 3)(2x + 1)\).

Step 1: Identify the Form

The expression is in the form of a quadratic equation, which is generally written as \( ax^2 + bx + c \). Here, \( a = 8 \), \( b = 10 \), and \( c = 3 \).

Step 2: Apply the AC Method

To factor the quadratic expression, we multiply \( a \) and \( c \): \( 8 \times 3 = 24 \). We need two numbers that multiply to 24 and add to 10, the middle coefficient 'b'.

Step 3: Find Suitable Numbers

The numbers 4 and 6 multiply to 24, and add to 10. So, these are our suitable numbers.

Step 4: Rewrite the Middle Term

Rewrite the middle term \( 10x \) as \( 4x + 6x \). This gives us: \[ 8x^2 + 4x + 6x + 3 \]

Step 5: Factor by Grouping

Group the terms: \[ (8x^2 + 4x) + (6x + 3) \] Factor out the greatest common factor from each group. From the first group, factor out \(4x\): \[ 4x(2x + 1) + 3(2x + 1) \] Here, \(2x + 1\) is common.

Step 6: Complete the Factorization

Factor out the common binomial \(2x + 1\): \[ (4x + 3)(2x + 1) \]. This is the completely factored form of the original expression.

Key Concepts

Quadratic Equation

Quadratic equations are fundamental in algebra and appear frequently in different mathematical problems. A quadratic equation can be expressed in the form of \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the sign of \( a \).

When you need to factor a quadratic expression, such as \( 8x^2 + 10x + 3 \), you're breaking it down into simpler terms that multiply together to give the original expression. This process makes solving quadratic equations easier.

Here, by identifying the coefficients, we recognize that \( a = 8 \), \( b = 10 \), and \( c = 3 \). These values are crucial for methods like factoring and solving the equation using the quadratic formula.

AC Method

The AC Method, also known as the "factor by splitting the middle term" method, is a reliable approach to factor quadratic equations when the leading coefficient \( a eq 1 \). This method helps determine two numbers that not only multiply to \( a \times c \) but also add up to \( b \), the middle term's coefficient.

In our example, with the quadratic expression \( 8x^2 + 10x + 3 \), we first multiply \( a \) and \( c \) which gives \( 8 \times 3 = 24 \). The goal is to find two numbers that multiply to 24 and add up to 10, derived from coefficient \( b \).

• The suitable numbers found are 4 and 6 because \( 4 \times 6 = 24 \) and \( 4 + 6 = 10 \).

By splitting 10x into 4x and 6x, we transform the expression in a way that facilitates easy grouping and further factoring.

Factoring by Grouping

Factoring by grouping is a practical technique used to simplify and factor expressions when the AC Method is applied. After determining the pair of numbers from the AC Method, you'll split the middle term of the quadratic equation.

Let's proceed by rewriting the quadratic expression \( 8x^2 + 10x + 3 \) as \( 8x^2 + 4x + 6x + 3 \). By grouping, you'll arrange the terms to facilitate factoring:

• Group the first two terms and the last two terms: \( (8x^2 + 4x) + (6x + 3) \)
• Factor out the greatest common factor (GCF) from each pair:
  
  • From \( 8x^2 + 4x \), the GCF is \( 4x \), giving us \( 4x(2x + 1) \).
  
  
  • From \( 6x + 3 \), the GCF is 3, leading to \( 3(2x + 1) \).

Now, both groups contain the common factor \( 2x + 1 \). By factoring this expression out, you get: \( (4x + 3)(2x + 1) \).
This fully factors the original quadratic expression, making it easier to solve or analyze.