Problem 109 - Algebra 2

Question

Factor each expression. $$ 4 x^{2}-8 x+3 $$

Step-by-Step Solution

Answer

The factored form of \(4x^2 - 8x + 3\) is \((2x - 1)^2\).

Step 1 - Identify the coefficients and the constant

The coefficients and the constant in this quadratic expression are 4, -8, and 3 respectively. These numbers are crucial in the process of factoring the quadratic expression.

Step 2 - Find two numbers

The challenge is to find two numbers that multiply to 12 (4 times 3) and add to -8. After trying a couple of combinations, it emerges that these numbers are -2 and -6 because (-2)(-6) equals 12 and -2 + (-6) equals -8.

Step 3 - Rewrite the middle term

The next step is to rewrite the middle term of the expression (-8x) using the two numbers that were found in the previous step. This results in \(4x^2 - 2x - 6x + 3\).

Step 4 - Factor by grouping

Factor the expression by grouping, i.e., put the first two terms together and the last two terms together and factor each group. This results in \(2x(2x - 1) - 1(2x -1)\). Then, factor out the common binomial term to get \((2x - 1)^2\).

Key Concepts

Quadratic Expressions

A quadratic expression is a mathematical equation of the form \(ax^2 + bx + c\). These expressions are characterized by having an \(x^2\) term.

The numerical values \(a\), \(b\), and \(c\) are known as coefficients. Each term in the expression signifies a specific part, such as:

• \(ax^2\): Quadratic term - determines the parabola's direction and width.
• \(bx\): Linear term - influences the slope and position of the parabola.
• \(c\): Constant term - shifts the parabola up or down.

Understanding these components helps you to manipulate and solve quadratic equations effectively.

Coefficient Identification

Identifying coefficients in a quadratic expression is one of the first steps in factoring it. You need to spot the values of \(a\), \(b\), and \(c\) easily.

In our given example \(4x^2 - 8x + 3\):

• The coefficient \(a\) is 4 - attached to \(x^2\).
• Coefficient \(b\) is -8 - found with \(x\).
• The constant term \(c\) is 3 - standing alone without \(x\).

Mastering this identification is crucial. It sets the stage for the next steps in factoring and solving.

Factoring by Grouping

Factoring by grouping is a method used to factor quadratic expressions, especially when a simple factorization is not evident. After rewriting the middle term, you group the terms to help see the factors more clearly.

In the expression \(4x^2 - 2x - 6x + 3\), group the first two terms and the last two terms:

• First group: \(4x^2 - 2x\)
• Second group: \(-6x + 3\)

Now, factor each group separately:

• \(2x(2x - 1) - 1(2x - 1)\)

Notice how \(2x - 1\) appears in both. This indicates that \(2x - 1\) is a common factor. Finish by factoring this out to get \((2x - 1)^2\). This technique helps neatly solve the expression by breaking it down into simpler parts.

Rewrite Middle Term

Rewriting the middle term is crucial for factoring complex quadratics where factoring cannot be easily done by inspection. To manage this, you first multiply the leading coefficient \(a\) and the constant term \(c\).

From the earlier example, multiplying \(4\) (leading coefficient) and \(3\) (constant) gives 12. Then, find two numbers that multiply to 12 and add up to -8, the middle term's coefficient.

After identifying \(-2\) and \(-6\) as such numbers, replace \(-8x\) with \(-2x\) and \(-6x\) in the expression, breaking it as \(4x^2 - 2x - 6x + 3\).

This step is about carefully transforming the middle term to enable easier grouping and factoring. It is an essential skill for solving tricky quadratic equations.