Problem 89 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The factorized form of the expression \(4x^2 - 8x + 4\) is \(4(x-1)^2\).

Step 1: Identify Common Factor

Notice the coefficients in our expression \(4x^2 - 8x + 4\). We see each term is divisible by 4. So, we extract the common factor 4.

Step 2: Factor out the Common Factor

The expression becomes \(4(x^2 - 2x + 1)\) after factoring out the common factor 4.

Step 3: Factor the Quadratic Equation

The expression inside the parenthesis \(x^2 - 2x + 1\) is a perfect square trinomial, and it can be factored into \((x-1)^2\).

Step 4: Write the Final Factored Form

As a result, our complete factorized form of the original expression \(4x^2 - 8x + 4\) is \(4(x-1)^2\).

Key Concepts

Common Factor

When dealing with expressions, finding a common factor is often the first step in simplifying or factoring them. A **common factor** is a number or an expression that divides each term of the expression evenly.

• To find a common factor, identify the largest number that can divide all coefficients of the terms in your expression.
• Look for common variables or expressions in the terms, and ensure the exponent is the smallest power present in all terms.
• Factoring out the common factor simplifies the expression and makes handling remaining terms easier.

In our original problem, the common factor is 4.
We factor it out to rewrite the expression as follows: \[4(x^2 - 2x + 1)\]This allows us to focus on the quadratic expression inside the parenthesis for further simplification.

Perfect Square Trinomial

A **perfect square trinomial** is a special type of quadratic expression. It results from squaring a binomial, and can be factored easily because of its structured form.

• The general form of a perfect square trinomial is \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\).
• It can be rewritten as \((a + b)^2\) or \((a - b)^2\) respectively.
• Identifying a perfect square trinomial is quick when you check if the first and last terms are perfect squares, and the middle term is twice the product of their square roots.

In the example, we have the expression \[x^2 - 2x + 1\]Here, \(x^2\) and \(1\) are perfect squares, and \(-2x\) is twice the product of \(x\) and \(-1\).
Thus, it factors to \[(x-1)^2\]Recognizing this special pattern saves time and simplifies factoring.

Quadratic Equation

A **quadratic equation** involves a polynomial of degree 2, typically expressed as \(ax^2 + bx + c = 0\). Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.

• Factoring involves rewriting the quadratic in terms of its roots with expressions like \((x - r_1)(x - r_2)\), where \(r_1\) and \(r_2\) are the roots.
• Perfect square trinomials simplify the factoring process as they can be immediately rewritten in squared form.
• Quadratic equations are central in algebra because they model numerous real-world phenomena.

In our case, once the common factor was extracted, we recognized the remaining expression as a quadratic in perfect square form: \[x^2 - 2x + 1 = (x-1)^2\]This understanding and manipulation of quadratics are essential for mastering algebraic techniques and problem-solving.