Problem 61 - Elementary Algebra

Question

For the following problems, factor, if possible, the trinomials. $$ x^{2}+10 x+25 $$

Step-by-Step Solution

Answer

Answer: The factored form of the trinomial \(x^2 + 10x + 25\) is \((x + 5)^2\).

Step 1: Identify the coefficient values

The given trinomial is \(x^2 + 10x + 25\), which can be written in the form \(ax^2 + bx + c\). Here, the coefficient values are: - \(a = 1\) - \(b = 10\) - \(c = 25\)

Step 2: Find two numbers whose product is \(ac\) and sum is \(b\)

We need to find two numbers whose product is equal to \(ac\) and whose sum is equal to \(b\). In our case, \(ac = 1 * 25 = 25\), and \(b = 10\). The two numbers are 5 and 5 since: - \(5 * 5 = 25\) - \(5 + 5 = 10\)

Step 3: Factor the trinomial

Now that we found the two numbers (5 and 5), we can factor the trinomial by writing it as the product of two binomials. Since both numbers are the same, we can write the trinomial as a perfect square: $$ x^2 + 10x + 25 = (x + 5)(x + 5) $$ You can also write it as: $$ x^2 + 10x + 25 = (x + 5)^2 $$ So, the factored form of the given trinomial is \((x + 5)^2\).

Key Concepts

Quadratic Expressions

A quadratic expression is a type of polynomial expression that includes a term with a variable squared. The standard form for a quadratic expression is written as \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. These expressions often appear in algebra and can model a variety of real-life phenomena, such as projectile motion.

• \(x^2\): The square term is what defines the expression as quadratic, contributing to its parabolic shape when graphed.
• \(bx\): The linear term, which influences the direction and position of the parabola.
• \(c\): The constant term, allowing for vertical adjustments of the graph.

Quadratic expressions can typically be factored or solved using methods like grouping, the quadratic formula, or completion of the square. Understanding their structure is key to unraveling complex mathematical problems.

Perfect Squares

Perfect squares arise when a number or expression is multiplied by itself. In algebra, a perfect square trinomial exhibits symmetry and can be factored into the square of a binomial. Identifying perfect squares simplifies factoring.

• If you have \( (a + b)^2 \), expanding this yields \( a^2 + 2ab + b^2 \), a common form of a perfect square trinomial.
• For our problem, \( x^2 + 10x + 25 \) is an example, as it can be rewritten as \((x + 5)^2\).

Being able to spot perfect square trinomials quickly can make factoring straightforward, helping to solve equations or simplify expressions efficiently.

Binomial Products

A binomial product results from multiplying two binomials. This can be done using the distributive property or the formula \( (a + b)(c + d) \).

• In the specific case of a trinomial \( x^2 + 10x + 25 \), the binomial product is \((x + 5)(x + 5)\).
• Rewriting it as \((x + 5)^2\) simplifies it further, showcasing a special type of binomial product called a square of a binomial.

The ability to recognize and manipulate binomial products is essential in algebra, simplifying complex problems and facilitating easier back-and-forth between different mathematical forms.