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Chapter 6: Problem 2

In Exercise \(1-4,\) completely factor each expression. $$x^{2}+4 x+4$$

Short Answer

Expert verified
The factored form of the expression is \((x+2)^2\).

Step by step solution

01

Identify the structure

The given expression \(x^{2}+4x+4\) is a quadratic trinomial because it is in the form \(ax^{2} + bx + c\). In this case, \(a=1, b=4, c=4\). Also, note that the trinomial has a structure of a perfect square trinomial \((x+d)^2=x^2+2dx+d^2\), where \(d^2=4\) and \(2d = 4\). So \(d=2\).
02

Factorize the expression

The quadratic equation can hence be written in the factored form as \((x+d)^2 = (x+2)^2\).

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Key Concepts

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

Perfect Square Trinomial
A perfect square trinomial is a special type of quadratic expression that can be written as the square of a binomial. Recognizing these trinomials is key to simplifying certain quadratic expressions. For example, the expression \(x^2 + 4x + 4\) is a perfect square trinomial.When identifying a perfect square trinomial, look for the format \((a+b)^2 = a^2 + 2ab + b^2\). Here, you can see:
  • \(a^2 = x^2\)
  • \(2ab = 4x\)
  • \(b^2 = 4\)
By matching these components from the expression, you determine \(a = x\) and \(b = 2\). This means that \(x^2 + 4x + 4\) can be written as \((x+2)^2\), illustrating its nature as a perfect square trinomial.
Factoring Quadratics
Factoring quadratics is the process of breaking down a quadratic equation into a product of its factors. This process simplifies solving equations and finding roots. A quadratic expression has the general form \(ax^2 + bx + c\). To factor, you often look for two numbers that multiply to \(c\) and add up to \(b\).For a perfect square trinomial like \(x^2 + 4x + 4\), factoring becomes straightforward once you recognize its structure. The factored form is the square of a binomial:
  • Identify the numbers that create a perfect square trinomial, such as \(b^2\) and \(2ab\).
  • Once identified, express the trinomial in its factored form \((x+2)^2\).
This process helps solve quadratic equations by making them easier to handle, especially when graphing or solving for roots.
Quadratic Equations
Quadratic equations are polynomial equations of degree two. They have the standard form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). They are essential in algebra as they appear in various problems from physics to economics.To solve quadratic equations, several methods can be used:
  • Factoring, if the equation can be easily rewritten as a product of factors.
  • Completing the square, which involves creating a perfect square trinomial.
  • The quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\), which provides solutions for any quadratic equation.
  • Graphing, where the solutions are the x-intercepts of the parabola.
Understanding different solving techniques is critical. Recognizing when the expression is a perfect square trinomial, as in \((x+2)^2 = 0\), helps directly find that \(x = -2\), providing a quick solution.

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Most popular questions from this chapter

In Exercises \(69-82,\) prove the given identities. $$\cos \left(\frac{\pi}{6}-x\right)=\cos \left(x-\frac{\pi}{6}\right)$$

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