Problem 34 Factor the perfect square trinom... [FREE SOLUTION]

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Elementary Algebra Within Reach

Ron Larson, Kimberly Nolting

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 6: Problem 34

Factor the perfect square trinomial. $$ a^{2}-12 a+36 $$

Short Answer

Expert verified

The factored form of the perfect square trinomial $a^2 - 12a + 36$ is $(a - 6)^2$.

Step by step solution

Identify Components of the Perfect Square Trinomial

In the trinomial $a^2 - 12a + 36$, $a^2$ is identified as 'a squared', -12a as '-2ab', and 36 as 'b squared'. From the -2ab component, we know that 2ab = -12a. Dividing both sides of the equation by 2a yields b = -6, which is the coefficient of a in the binomial.

Factor the Trinomial

To factor the trinomial, take the square root of the first term ($\sqrt{a^2} = a$) and the last term ($\sqrt{36} = 6$). Now, sum up the square roots with the symbol in front of the b squared term, resulting in $a - 6$. So the trinomial can be factored into (a - 6)^2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Square Trinomial

Perfect square trinomials are expressions that can be rewritten as the square of a binomial. They take the form $x^2 + 2xy + y^2$ or $x^2 - 2xy + y^2$. When factoring these trinomials, we're essentially identifying what squared binomial results in the given trinomial. In its simplest terms, a perfect square trinomial will have:

The square of a term, like $x^2$, as its first part.
The square of another term, like $y^2$, as its last component.
The middle term that represents twice the product of the two terms, such as $2xy$.

To factor, find the square root of the first and the last terms, and use these to create a binomial squared. For example, $a^2 - 12a + 36$ is recognized as a perfect square trinomial because it can be written as $(a - 6)^2$, where $a$ and $6$ are the terms being squared.

Algebraic Expressions

Algebraic expressions involve variables, constants, and operations (like addition, subtraction, and multiplication). These expressions are central in algebra and are used to represent mathematical relationships. An expression can be as simple as a single number or letter (e.g., $x$ or $5$), or it may involve multiple terms (e.g., $3x^2 - 4x + 7$).

An algebraic expression does not have an equal sign, distinguishing it from an equation.
Terms in an expression are often combined using addition or subtraction.
Each term can be a constant, a variable, or a product/quotient involving constants and variables.

In the expression $a^2 - 12a + 36$, the components $a^2$, $-12a$, and $36$ represent different terms, combined to form a trinomial, which is a type of algebraic expression featuring three terms.

Quadratic Equations

Quadratic equations are polynomial equations of degree 2, and they are typically written in the form $ax^2 + bx + c = 0$, where $a, b,$ and $c$ are constants, and $a eq 0$. These equations are prevalent in algebra due to their applications in various fields like physics, engineering, and economics.
To solve quadratic equations, you might use methods such as:

Factoring: Break down the quadratic expression into two binomial expressions.
Completing the square: Rewriting the equation in the form $(x+p)^2 = q$.
Using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$.

In the example of the perfect square trinomial, $a^2 - 12a + 36$, even though it is not set equal to zero, it can be factored as $(a-6)^2$ and set in an equation like $(a-6)^2 = 0$ to find that $a = 6$. This connection highlights how factoring is applied within quadratic equations.

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91影视

Factor the perfect square trinomial. $$ a^{2}-12 a+36 $$

Short Answer

Step by step solution

Identify Components of the Perfect Square Trinomial

Factor the Trinomial

Key Concepts

Perfect Square Trinomial

Algebraic Expressions

Quadratic Equations

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