Problem 72 - Algebra 2

Question

State whether each trinomial is a perfect square. If so, factor it. \(36 x^{2}-60 x+25\)

Step-by-Step Solution

Answer

Yes, it is a perfect square. The factorization is \((6x - 5)^2\).

Step 1: Identify the Structure

A trinomial is a perfect square if it has the form \[a^2 - 2ab + b^2\] or \[a^2 + 2ab + b^2\] where: - The first term is a square (\(a^2\)) - The last term is a square (\(b^2\)) - The middle term is twice the product of the square roots of the first and last terms (\(2ab\)). Let's identify \(a\) and \(b\) in our trinomial.

Step 2: Analyze the First Term

The first term is \(36x^2\), which is \((6x)^2\). Thus, \(a = 6x\).

Step 3: Analyze the Last Term

The last term is 25, which is \(5^2\). Thus, \(b = 5\).

Step 4: Check the Middle Term

The middle term should be \(-2ab\) or \(2ab\). Let's calculate: \[-2 imes 6x imes 5 = -60x\]. This matches the middle term \(-60x\) in the trinomial.

Step 5: Conclusion on Perfect Square

Since the first term \((6x)^2\), the last term \(5^2\), and the middle term \(-60x\) satisfy the form \((a - b)^2 = a^2 - 2ab + b^2\), the trinomial is a perfect square.

Step 6: Factor the Trinomial

The trinomial \(36x^2 - 60x + 25\) factors as \((6x - 5)^2\).

Key Concepts

Trinomial Factoring

Trinomial factoring is a process in algebra that involves breaking down a trinomial into simpler components. A trinomial is a type of polynomial that contains three terms. It often looks like this: \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is a variable. The goal is to express the trinomial as a product of simpler expressions. Factoring trinomial expressions can help solve equations and simplify them.

• First, identify the standard form \(ax^2 + bx + c\).
• Check if the trinomial is a perfect square, meaning it can be expressed as \((a + b)^2\) or \((a - b)^2\).
• Examine each term to see if it fits into a recognizable pattern, such as \(a^2 - 2ab + b^2\).

Understanding these elements helps in finding the factors, streamlining complex algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the core of much of algebra. In the context of trinomials, you deal with polynomial expressions where variables are raised to powers, combined with constants, and separated by addition or subtraction symbols.An expression like \(3x^2 + 4x - 7\) includes:

• Variables: Letters that represent numbers, such as \(x\).
• Coefficients: Numbers multiplying the variables, like 3 and 4 in the expression.
• Constants: Numbers that stand alone, such as -7.

Working with algebraic expressions involves performing operations like addition, subtraction, multiplication, and factoring. Understanding how to manipulate these building blocks is crucial for solving algebraic problems effectively.

Factoring Techniques

Factoring techniques are methods used to decompose algebraic expressions into simpler terms. These methods are important in solving equations, simplifying expressions, or finding zeroes of polynomials. For trinomials suspected to be perfect squares, a specific technique is applied:

• Identify squared terms. Recognize patterns such as \(a^2 - 2ab + b^2\).
• Calculate roots. Find square roots of the first and last terms to determine possible factors.
• Match the middle term. Check if the middle term equals \(2ab\) or \(-2ab\).

These steps guide you in determining if a perfect square exists and allow you to express the trinomial as a square of a binomial, simplifying the expression to something like \((a-b)^2\) or \((a+b)^2\).

Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). These equations are foundational in algebra and frequently appear in mathematical modeling and real-world problem-solving. Techniques such as factoring are pivotal in solving them.For a perfect square trinomial like \(36x^2 - 60x + 25\), if recognized early, it can be rewritten and solved as a factored expression, \((6x-5)^2 = 0\). This implies

• Setting \(6x-5 = 0\), solve for \(x\).
• Find the solution, which is \(x = \frac{5}{6}\).

These solved factors provide roots of the quadratic equation. Mastering factorization not only gives solutions quickly but also makes the process more efficient, providing deeper insight into the structure of quadratics.