Problem 27 - Algebra 1: Concepts and Skills

Question

PERFECT SQUARES Factor the expression. $$ x^{2}+8 x+16 $$

Step-by-Step Solution

Answer

The factored form of the expression \(x^2+8x+16\) is \((x+4)^2\).

Step 1: Identifying the Form

First and foremost, confirm whether the given quadratic equation fits the form of a perfect square trinomial. For that, identify \(a^2\) as \(x^2\) and \(b^2\) as 16. From this we get \(a\) as \(x\) and \(b\) as \(4\). Evaluate \(2ab\) to be \(2*4*x = 8x\), which matches the middle term of the given expression.

Step 2: Factoring the Expression

After confirming that the expression takes the form of a perfect square trinomial, factor it. By the nature of perfect square trinomials, the factored form of \(x^2 + 8x + 16\) should be \((x+4)^2\).

Step 3: Verifying the Factorization

Finally, to 100% confirm that the factoring is correct, expand \((x+4)^2\) to get back the original quadratic equation. Applying the \(FOIL\) method, \((x+4)^2\) becomes \(x^2 + 2*4*x + 16 = x^2 + 8x + 16\), which matches the original expression.

Key Concepts

Factoring Expressions

Factoring expressions is an essential skill in algebra, especially when dealing with quadratic equations. It involves rewriting a complex expression as a product of simpler ones, typically to make solving equations easier or to simplify expressions. The most common expressions students learn to factor are linear and quadratic expressions.

Focusing on quadratic expressions, which are polynomials of the form \( ax^2 + bx + c \), one special variation we encounter is the perfect square trinomial. This occurs when \(a\) and \(b\) can be written as \(a^2\) and \(2ab\), respectively, with \(b^2\) being the last term. For example, \(x^2 + 8x + 16\) is factored as \( (x+4)^2 \). Here’s why it's a perfect square: it has the form \(a^2 + 2ab + b^2\), where \(a = x\) and \(b = 4\). Factoring expressions like this requires recognizing these patterns and applying them correctly.

Quadratic Equations

Quadratic equations play a vital role in mathematics and real-world problem-solving. They are second-degree polynomial equations with a standard form of \( ax^2 + bx + c = 0 \). The solutions to these equations, also known as the roots, can be found using various methods, including factoring, completing the square, and the quadratic formula.

In the context of perfect square trinomials, like \( x^2 + 8x + 16 = 0\), recognizing that the equation represents a perfect square can immediately tell us its solution without needing further techniques. Factoring the expression gives us \( (x+4)^2 = 0 \), and taking the square root indicates that \(x = -4\) is the only solution. This method simplifies the process and is especially useful when equations are designed to be perfect squares.

FOIL Method

The FOIL method stands for First, Outer, Inner, Last. It’s a technique used to multiply two binomials. FOIL is helpful in both factoring and expanding expressions, and understanding it is crucial for solving a wide range of algebraic problems.

For instance, to verify the factored form of the perfect square trinomial factored in our example, we can use the FOIL method to expand \( (x+4)^2 \). Here’s how you FOIL it out: Multiply the First terms (\(x * x = x^2\)), then the Outer (\(x * 4 = 4x\)), followed by the Inner (\(4 * x = 4x\)), and finally, the Last terms (\(4 * 4 = 16\)). Adding these together, \(x^2 + 4x + 4x + 16\), condenses to \(x^2 + 8x + 16\), which matches our original expression. Learning to FOIL effectively can greatly aid in both expanding and factoring expressions, providing a solid foundation for algebraic manipulation.