Problem 55 - Calculus: One and Several Variables

Question

Write the expression in factored form. \(4 x^{2}+12 x+9\).

Step-by-Step Solution

Answer

The factored form of the expression \(4 x^{2}+12 x+9\) is \((2x + 3)^{2}\).

Step 1: Identify a, b and c

In the equation \(4 x^{2}+12 x+9\), we can identify that \(a\) is \(2x\), \(b\) is 3 and \(c\) is also 3. As the equation is already in a simplified form, we can confirm that it is a perfect square trinomial because \(a^{2} = 4x^{2}\), \(2ab = 12x\) and \(b^{2} = 9\).

Step 2: Factor the quadratic expression

We start factoring by rewriting the quadratic expression in factored form. As per the standard form for perfect square trinomials, it can be written as \((a+b)^{2}\). Here, \(a\) is \(2x\) and \(b\) is 3, so our factored form will be \((2x + 3)^{2}\). Hence, \(4 x^{2}+12 x+9\) can be factored as \((2x + 3)^{2}\).

Key Concepts

Understanding Perfect Square Trinomials

A perfect square trinomial is a special type of quadratic expression that takes the form \(a^2 + 2ab + b^2\). This trinomial can always be rewritten as the square of a binomial. In simpler terms, it's an expression that looks like a settled little package or a wrapped-up gift. Recognizing a perfect square trinomial is key when simplifying quadratic expressions.

• The first term \(a^2\) is the square of the first term \(a\) in the binomial.
• The second term \(2ab\) is twice the product of the two terms.
• The third term \(b^2\) is the square of the last term \(b\) in the binomial.

If the given trinomial matches this pattern, then it can neatly be arranged as \((a + b)^2\). This means your job becomes much easier when dealing with factoring, as you essentially "unwrap" the trinomial into its simpler binomial square form.

Exploring Quadratic Expressions

Quadratic expressions are algebraic expressions characterized by variables raised to the second power. They commonly take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable.Quadratics can be transformed and solved through various methods like factoring, using the quadratic formula, or completing the square. In context with our exercise, we focus on recognizing specific patterns that help in factoring more efficiently.

• Identifying whether a quadratic is a perfect square trinomial can simplify the process.
• Knowing which method to use for different types of quadratics is crucial.
• Graphically, quadratic equations form a parabola, which can give insights into their roots or solutions.

When approached methodically, quadratics reveal their structure, allowing easier handling whether you're performing algebraic operations or solving them for roots.

The Process of Algebraic Factoring

Algebraic factoring involves breaking down complex expressions into simpler components or the products of their factors. It's like reducing a recipe to its individual ingredients. The main goal is to express the quadratic as a product of binomials, which simplifies both the expression and the process of solving the equation.

• Identifying a perfect square trinomial simplifies the factoring process into recognizing the binomial square form.
• Other general methods include finding two numbers that multiply to give the product \(ac\) (the coefficient of \(a\times c\) in the quadratic formulation) and add to the middle coefficient \(b\).
• Factoring transforms the expression into a product of simpler expressions that, when multiplied, return the original quadratic.

Factoring is akin to looking for a pattern or a formula hidden within the quadratic. When performed correctly, this method not only provides a solution efficiently but also ensures understanding of the quadratic's structure and the relationship between its terms.