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Chapter 10: Problem 21

Factor. $$9+24 a+16 a^{2}$$

Short Answer

Expert verified
The factored form of the trinomial is \((4a + 3)^2\).

Step by step solution

01

Rearrange the trinomial

Rearrange the trinomial in the form \(ax^2 + bx + c\). It becomes \(16a^2 + 24a + 9\).
02

Identify if the trinomial is a perfect square

Compare the trinomial with the form of a perfect square trinomial. \(b^2 = 9\) gives \(b = \pm 3\). \(a = 4\) returns \(2ab = 24a\) with \(b = 3\). Hence, the trinomial is a perfect square trinomial.
03

Factoring the trinomial

Factor the trinomial using the perfect square trinomial formula. The factored form will be \((4a + 3)^2\).

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Key Concepts

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

Perfect Square Trinomials
Perfect square trinomials are special types of polynomials that can be expressed in the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). These trinomials emerge when a binomial is squared. For example, \( (a + b)^2 = a^2 + 2ab + b^2 \). Recognizing these patterns makes factoring much easier because we only need to identify the terms that complete the square.
  • Identify the first and last terms as squares of some expressions.
  • Check if the middle term (the one with \(ab\) or \(-ab\)) fits the square pattern.
Finding a match confirms it's a perfect square trinomial, simplifying the factoring process into a single binomial squared.
Algebraic Expressions
Algebraic expressions consist of numbers, variables, and arithmetic operations. These are vital in mathematics, as they allow us to describe relationships and solve equations. In the context of factoring trinomials, understanding algebraic expressions helps us identify how to rearrange and simplify them.
When working with expressions like \(9 + 24a + 16a^2\), it鈥檚 essential to;
  • Recognize variables and constants.
  • Identify linked terms through like variables.
Rearranging expressions into a standard quadratic form \(ax^2 + bx + c\) facilitates solving by revealing underlying patterns, like those seen in perfect square trinomials.
Quadratic Equations
Quadratic equations, usually expressed as \(ax^2 + bx + c = 0\), involve terms where the highest power of the unknown variable is 2. Solving these equations is fundamental in algebra, as they have numerous applications.
In factoring trinomials like \(16a^2 + 24a + 9\), recognizing it as a part of a quadratic equation \(ax^2 + bx + c\) enables us to apply various methods to find solutions; ultimately, trying to express it in factored form. Comparing and manipulating the equation reveals its properties, such as symmetry and solutions represented by x-intercepts.
  • Understand the structure: leading coefficient, middle term, and constant.
  • Utilize factors like perfect square trinomials to find simplified forms.
This approach simplifies finding solutions, like factored expressions of quadratic equations.

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Most popular questions from this chapter

Factor by grouping. $$60 x^{2}+95 x+20$$

For Exercises 78 to \(131,\) factor completely. $$2 t^{2}-24 t s+70 s^{2}$$

For Exercises 78 to \(131,\) factor completely. $$y^{2}+4 y z-21 z^{2}$$

Factor by grouping. $$12+11 x-x^{2}$$

For Exercises 78 to \(131,\) factor completely. $$a^{2}-13 a b+42 b^{2}$$
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