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Chapter 6: Problem 122

Factor completely. \(p^{2}+p+\frac{1}{4}\)

Short Answer

Expert verified
The expression factors to \( \left( p + \frac{1}{2} \right)^2 \).

Step by step solution

01

Identify the Quadratic Expression

The given expression is a quadratic expression: \( p^2 + p + \frac{1}{4} \). This has one variable, \( p \), and is in standard form: \( ax^2 + bx + c \), where \( a = 1 \), \( b = 1 \), and \( c = \frac{1}{4} \).
02

Check for Perfect Square Trinomial Pattern

A perfect square trinomial has the form \( (x + n)^2 = x^2 + 2nx + n^2 \). First, note that \( 2n = b = 1 \), giving \( n = \frac{1}{2} \). The term \( n^2 \) needs to equal \( c = \frac{1}{4} \), and indeed it does since \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \).
03

Rewrite as a Perfect Square

Since the quadratic is a perfect square trinomial, rewrite it as the square of a binomial: \( p^2 + p + \frac{1}{4} = \left(p + \frac{1}{2}\right)^2 \).
04

Final Verification

Verify that the expression \( (p + \frac{1}{2})^2 = p^2 + p + \frac{1}{4} \) by expanding \( (p + \frac{1}{2})(p + \frac{1}{2}) \) back to ensure it matches the original expression.

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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 which can be conveniently factored into the square of a binomial. This means it can be written in a form that looks like \[ (x + n)^2 = x^2 + 2nx + n^2 \]. This pattern is handy because it simplifies the process of factoring and makes calculations easier.

To recognize a perfect square trinomial, look for:
  • The first term as a perfect square. In our example, \( p^2 \) is a perfect square as it is equal to \( (p)^2 \).
  • The last term as \( n^2 \), which should also be a perfect square. Here, \( \frac{1}{4} \) is \( \left(\frac{1}{2}\right)^2 \).
  • The middle term should be twice the product of the square roots of the first and last terms, i.e., \( 2n \). In this equation, it's \( 1 \), which is twice \( \frac{1}{2} \).
Identifying this pattern enables one to quickly rewrite the quadratic as the square of a binomial thus saving time and effort in more complex equations.
Quadratic Expression
A quadratic expression is a polynomial that includes a squared variable term. Typically written in general form as \[ ax^2 + bx + c \], it features three elements:
  • The squared term \( ax^2 \) which holds the highest power, making it quadratic.
  • The linear term \( bx \) which is the first power.
  • A constant term \( c \) which is independent of the variable.
In the problem we're addressing, the quadratic expression is \( p^2 + p + \frac{1}{4} \) where \( a = 1 \), \( b = 1 \), and \( c = \frac{1}{4} \). Recognizing the parts of a quadratic expression helps in determining appropriate factoring techniques and understanding the behavior of its graph as a parabola.
Standard Form of a Quadratic Equation
The standard form of a quadratic equation is expressed as \[ ax^2 + bx + c = 0 \], where \( a \), \( b \), and \( c \) are coefficients and constants. This form is largely used due to its straightforward structure, making further analysis easier.

To solve problems using this form:
  • Identify and assign values to \( a \), \( b \), and \( c \). This aids in simplifying the expression.
  • When factoring, look for patterns such as perfect square trinomials or use formulas like the quadratic equation \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \).
Standard form helps verify the nature of roots and solutions for the equation. It provides a basis that simplifies solving and interpreting quadratic equations.

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Most popular questions from this chapter

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ u^{2}-18 u+81 $$

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 8 x^{4}-8 $$

Solve each equation. $$ 2 a^{3}+a^{2}-32 a-16=0 $$

For each of the following polynomials, which factoring method would you use first? $$ x^{2}+18 x+81 $$

A doctor advises a patient to exercise at least 15 minutes but less than 30 minutes per day. Use a compound inequality to express the range of these times \(t\) in minutes.
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