/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 36 Factor the perfect squares. $$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 36

Factor the perfect squares. $$ 25 x^{2}+10 x+1 $$

Short Answer

Expert verified
\[ 25x^{2} + 10x + 1 = (5x + 1)^{2} \]

Step by step solution

01

- Identify the quadratic expression

Recognize that the quadratic expression given is a trinomial: oindent \[ 25x^{2} + 10x + 1 \]
02

- Recognize the perfect square form

Identify that the expression is a perfect square trinomial. A perfect square trinomial follows the form: oindent \[ a^{2} + 2ab + b^{2} = (a + b)^{2} \].Here, we compare oindent \[ 25x^{2} + 10x + 1 \] with oindent \[ a^{2} + 2ab + b^{2} \] to find values of oindent \[ a \] and oindent \[ b \].
03

- Determine values for a and b

Find oindent \[ a \] and oindent \[ b \] from the expression. oindent For the first term oindent \[ 25x^2 \]: oindent \[ a^2 = 25x^2 \], so oindent \[ a = 5x \]. For the last term oindent \[ 1 \]: oindent \[ b^2 = 1 \], so oindent \[ b = 1 \].
04

- Verify the middle term

Check the middle term oindent \[ 10x \]: oindent \[ 2ab = 2(5x)(1) = 10x \]. It matches the middle term from the original expression, confirming our values of oindent \[ a = 5x \] and oindent \[ b = 1 \].
05

- Write the factored form

Now write the factored form using oindent \[ a \] and oindent \[ b \] in the formula: oindent \[ (a + b)^{2} \]. Substitute oindent \[ a = 5x \] and oindent \[ b = 1 \] into the equation: oindent \[ (5x + 1)^{2} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

quadratic expression
A quadratic expression is a polynomial where the highest degree is two. This means the term with the highest power of the variable is squared. Generally, it takes the form: \[ ax^2 + bx + c \]. Here, \( a \), \( b \), and \( c \) are constants and \( x \) is the variable. Quadratic expressions can be factored, solved, and analyzed in various ways, but one common approach is factoring, especially when the expression forms a perfect square trinomial.
perfect square trinomial
A perfect square trinomial is a specific type of quadratic expression that can be written as the square of a binomial. It follows the form: \[ a^{2} + 2ab + b^{2} = (a + b)^{2} \]. Identifying perfect square trinomials is crucial because they simplify factoring significantly. To identify a perfect square trinomial, check:
  • If the first and last terms are perfect squares.
  • If the middle term is twice the product of the square roots of the first and last terms.
For example, in the expression \( 25x^2 + 10x + 1 \):
  • First term: \( 25x^2 = (5x)^2 \)
  • Last term: \( 1 = (1)^2 \)
  • Middle term: \( 10x = 2 \times 5x \times 1 \)
Since all conditions match, the expression is a perfect square trinomial.
factored form
The factored form of a perfect square trinomial is achieved by rewriting it as the square of a binomial. Using the example \( 25x^2 + 10x + 1 \):
  • First, recognize it as a perfect square trinomial.
  • Then, use the identified values of \( a \) and \( b \).
  • Substitute these into the binomial form.
In this case:
  • \( a = 5x \)
  • \( b = 1 \)
So, the factored form is \( (5x + 1)^2 \). Factoring simplifies the expression, making it easier to solve equations or understand the properties of the quadratic function it represents.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{x+h}-\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}$$

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$ \frac{\left(16 x^{2} y^{-1 / 3}\right)^{3 / 4}}{\left(x y^{2}\right)^{1 / 4}} $$

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{2-\sqrt{5}}{2+3 \sqrt{5}}$$

Simplify each expression. Assume that all variables are positive when they appear. $$6 \sqrt{5}-4 \sqrt{5}$$

Simplify each expression. Assume that all variables are positive when they appear. $$(\sqrt[3]{3} \sqrt{10})^{4}$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.