Problem 23 - Algebra and Trigonometry

Question

17–24 ? Use a Factoring Formula to factor the expression. $$ x^{2}+12 x+36 $$

Step-by-Step Solution

Answer

The expression \(x^2 + 12x + 36\) factors to \((x+6)^2\).

Step 1: Identify the expression

The given expression is \(x^2 + 12x + 36\). This is a quadratic expression in the form of \(ax^2 + bx + c\). Here, \(a = 1\), \(b = 12\), and \(c = 36\).

Step 2: Recognize it as a perfect square trinomial

A perfect square trinomial has the form \((x + d)^2 = x^2 + 2dx + d^2\). In our expression, we want to check if it fits this pattern. Here, \(36\) is a perfect square \((6^2 = 36)\) and consequently, \(12 = 2 \times 6\), which fits the pattern \(2d = 12\).

Step 3: Confirm and factor using the formula

Since the conditions for a perfect square trinomial are met (\(b = 2d\) and \(c = d^2\)), the expression \(x^2 + 12x + 36\) can be factored as \((x + 6)^2\). Therefore, the factored form is \((x + 6)(x + 6)\).

Key Concepts

Perfect Square Trinomials

Perfect square trinomials are a special type of quadratic expression that can be factored into a binomial squared. They follow a specific pattern that makes them easy to recognize and factor. This pattern is:

• \[ (x + d)^2 = x^2 + 2dx + d^2 \]

In this pattern, "d" is a number that, when squared, equals the constant term "c" in the quadratic expression. The linear term "b" is exactly twice the value of "d". Recognizing this pattern allows you to quickly factor the trinomial.

For example, in the expression \(x^2 + 12x + 36\), notice that 36 is a perfect square since it equals \(6^2\), and 12 is two times 6, matching the pattern of a perfect square trinomial: \(b = 2d\). Therefore, it can be factored as \((x + 6)^2\). This ability to rearrange quadratics into a binomial squared simplifies solving quadratic equations and understanding the graph of a parabola.

Quadratic Expressions

A quadratic expression is an algebraic expression of degree 2, meaning the highest exponent of the variable is 2. It follows the general form:

• \[ ax^2 + bx + c \], where
• "a", "b", and "c" are constants with "a" not equal to zero.

Quadratics are foundational in algebra because they represent a wide range of phenomena and have versatile applications from physics to finance.

The shape this expression forms when graphed is known as a parabola. The term "ax^2" determines the parabola's direction and width, while "bx" and "c" adjust its position on the graph. Recognizing how these expressions behave allows you to apply different factoring techniques, such as finding roots via the quadratic formula or completing the square. For example, recognizing our exercise expression \(x^2 + 12x + 36\) fits this quadratic pattern is the first step in understanding how to manipulate and solve it.

Factoring Formulas

Factoring formulas are specific methods used to decompose expressions into simpler multipliers or factors. They simplify expressions and solve equations. In the context of quadratic expressions, several commonly used formulas exist:

• Perfect Square Trinomial: \[ (x + d)^2 = x^2 + 2dx + d^2 \]
• Difference of Squares: \[ a^2 - b^2 = (a + b)(a - b) \]
• Quadratic Formula: This is used when simple factoring isn't feasible: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For our expression \(x^2 + 12x + 36\), we used the perfect square trinomial formula. Recognizing such patterns allows you to break down complex expressions and solve equations more easily. Factoring is not just a technique but a tool to uncover solutions and understand relationships between algebraic expressions.