Problem 66 - Algebra 1

Question

Completely factor the expression. $$4 x^{2}-28 x+49$$

Step-by-Step Solution

Answer

The completely factored form of \(4x^{2} - 28x + 49\) is \((2x - 7)^{2}\).

Step 1: Identify the coefficients

First identify the coefficients in the quadratic equation \(4x^{2} - 28x + 49\). The coefficient a is attached to \(x^{2}\) which is 4, the coefficient b is attached to x which is -28 and the constant c is 49.

Step 2: Factor

Then factor the quadratic equation. This can be done by observing the pattern and making use of the identity \((a-b)^{2} = a^{2} - 2ab + b^{2}\). So the given expression can be rewritten as: \((2x - 7)^{2}\) as \(2^{2} = 4\) and \(2*2*7 = 28\) and \(7^{2} = 49\)

Step 3: Write the final factored form

The completely factored form of the given quadratic expression \(4x^{2} - 28x + 49\) is \((2x - 7)^{2}\).

Key Concepts

Quadratic Expression

A quadratic expression is a polynomial of degree two, which generally takes the form \( ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). Quadratic expressions typically feature a variable raised to the power of two, that is, \( x^2 \).

In our exercise example, the quadratic expression is \( 4x^2 - 28x + 49 \). This expression is second-degree because the highest power of the variable \( x \) is two. To handle these expressions, especially when factoring, understanding the roles of the coefficients \( a \), \( b \), and \( c \) in forming the parabola when graphed is important.

Identifying these coefficients is always the first step in solving or factoring quadratics, as it helps recognize patterns or opportunities to simplify the expression further, often into forms such as perfect square trinomials.

Perfect Square Trinomial

Perfect square trinomials are special types of trinomial expressions that can be rewritten as the square of a binomial. They generally follow the pattern \( a^2 - 2ab + b^2 \) or \( a^2 + 2ab + b^2 \). These patterns simplify factoring because they represent expressions that are squares of sums or differences of two terms.

Looking at the quadratic expression \( 4x^2 - 28x + 49 \), we observe it fits the pattern of \( (a - b)^2 = a^2 - 2ab + b^2 \).

The expression matches because:

• \( a^2 = (2x)^2 = 4x^2 \),
• \( 2ab = 2 \times 2x \times 7 = 28x \),
• and \( b^2 = 7^2 = 49 \).

This implies the expression is actually \((2x - 7)^2\), showcasing it as a perfect square trinomial.

Recognizing perfect square trinomials allows for efficient factoring and is a valuable method for simplifying quadratic expressions in algebra.

Factoring Method

Factoring is the process of breaking down an expression into simpler terms or factors that, when multiplied together, give back the original expression. Factoring is a crucial skill in algebra, especially for solving quadratic equations.

The factoring method we use is influenced by patterns, such as recognizing perfect square trinomials. For \( 4x^2 - 28x + 49 \), we factored it by recognizing the pattern and rewriting it as \((2x - 7)^2\).

Here's a quick guide on factoring quadratics:

• Check for a greatest common factor (GCF) first. If possible, factor it out.
• Identify if the expression fits recognizable patterns like perfect square trinomials or difference of squares.
• Attempt to express the quadratic in simpler binomial terms that, when squared, give the original expression.

In our example, once identified as a perfect square, \( 4x^2 - 28x + 49 \) is factored to \((2x - 7)^2\). This efficiency in recognizing patterns simplifies problems, making it critical for math success. By mastering these factoring techniques, tackling quadratics becomes more straightforward, allowing one to solve or simplify equations with ease.