Problem 99 - Introductory Algebra for College Students

Question

Will help you prepare for the material covered in the next section. $$\text { Factor: } x^{2}+8 x+16$$

Step-by-Step Solution

Answer

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

Step 1: Identify the equation

The quadratic equation given is: \(x^2 + 8x + 16\). It's in the standard form \(a^2 + 2ab + b^2\). Here, \(a = x\) and \(b = 4\) (since \(4^2 = 16\)).

Step 2: Apply the formula for factoring

To use the perfect square trinomial factoring formula \((a + b)^2 = a^2 + 2ab + b^2\), replace 'a' with 'x' and 'b' with '4'. The factored form will be \((a + b)^2\).

Step 3: Write the final factored form

Inserting our values into the formula, we get \((x+4)^2\). So, the factored form of the given equation \(x^2 + 8x + 16\) is \((x+4)^2\).

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two. It generally has the form \(ax^2 + bx + c = 0\). The quadratic equation is a cornerstone of algebra, representing equations where the highest power of the variable \(x\) is two. It can manifest in various forms, but recognizing its standard form helps in solving and factoring it easily.

Key parts of the quadratic equation include:

• The "\(x^2\)" term is the quadratic term and it determines the parabola's shape.
• The "\(bx\)" term is the linear term, influencing the parabola's slope and direction.
• The "\(c\)" term is the constant and shifts the graph up or down without affecting its shape.

Understanding these components helps in manipulating the equation for solving or factoring purposes.

Perfect Square Trinomials Explained

Perfect square trinomials are special types of quadratic equations formed by squaring a binomial. In the standard form, they look like \(a^2 + 2ab + b^2\). Recognizing a perfect square trinomial can simplify the factoring process significantly, allowing you to write the expression in a compact form.

Characteristics of a perfect square trinomial:

• The first and last terms are perfect squares.
• The middle term equals twice the product of the square roots of the first and last terms.

For example, \(x^2 + 8x + 16\):

• The first term "\(x^2\)" is the square of \(x\).
• The last term "16" is the square of 4.
• The middle term "8x" is twice the product of \(x\) and 4.

Hence, \(x^2 + 8x + 16\) can be factored as \((x + 4)^2\). This understanding streamlines the process, especially when dealing with quadratic equations that look more complex.

Factoring Methods for Quadratic Equations

Factoring is a method used to express a quadratic equation as a product of its binomials. There are several approaches to factoring, but for perfect square trinomials, we use a specific method due to its simplicity and efficiency.

Steps for factoring a perfect square trinomial:

• Write the trinomial in the standard form \(a^2 + 2ab + b^2\).
• Identify \(a\) and \(b\) based on the square roots of the first and last terms.
• Apply the formula \((a + b)^2\) or \((a - b)^2\) depending on the sign of the middle term.

For reference, if given the trinomial \(x^2 + 8x + 16\), we identify that \(a = x\) and \(b = 4\), thus the factored form is \((x + 4)^2\).

Mastering these factoring methods not only aids in solving quadratic equations but also paves the way for tackling higher-degree polynomials. It's an essential tool for algebra students.