/*! 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 88 Complete the square of \(y^{2}+3... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 88

Complete the square of \(y^{2}+3 y\) and write the result as the square of a binomial. [1.1]

Short Answer

Expert verified
The square of a binomial of the given expression \(y^{2}+3 y\) is \((y+\frac{3}{2})^{2}\).

Step by step solution

01

Identify the pattern

Notice that the expression \(y^{2}+3y\) is missing its constant term. We know that a perfect square trinomial is in the form \( (a+b)^{2}= a^{2} + 2ab + b^{2} \). In this case, we have \(a^{2}\) and \(2ab\) (with \(a=y\) and \(b=\frac{3}{2}\)), and we must find \(b^{2}\).
02

Complete the square

To match the perfect square trinomial pattern, we can add \(b^{2}\) to our original expression. For our case, we calculate \(b^{2}\) as follows: \(b^{2} = (\frac{3}{2})^{2} = \frac{9}{4}\). So, our perfect square trinomial is: \(y^{2}+3y + \frac{9}{4}\).
03

Express as a square of a binomial

Now that we have our perfect square trinomial, it should match the format \((a+b)^{2}\). Here, \(a=y\) and \(b=\frac{3}{2}\). Therefore, our expression becomes: \((y+\frac{3}{2})^{2}\)

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.

Perfect Square Trinomial
Understanding perfect square trinomials is crucial when completing the square in quadratic expressions. A perfect square trinomial can be identified by its general form, \( (a+b)^2 = a^2 + 2ab + b^2 \). In this expression, the first term \( a^2 \) and the third term \( b^2 \) are squares, while the middle term \( 2ab \) is twice the product of the roots \( a \) and \( b \).
Breaking this down:
  • \( a \) is the square root of the first term.
  • \( b \) is needed to ensure that the middle term is twice the product of \( a \) and \( b \).
To convert any quadratic expression into a perfect square trinomial, we find \( b^2 \), which is missing, and add it to the expression. This adjustment creates the pattern needed to rewrite the expression as a binomial square.
Binomial Square
Once we have a perfect square trinomial, expressing it as a binomial square is the next straightforward step. A binomial square is written in the form \((a + b)^2\).
The process involves identifying the values of \( a \) and \( b \) that make up the terms of the trinomial:
  • \( a \) usually represents the variable term's coefficient divided by the corresponding multiplier from the perfect square trinomial formula.
  • \( b \), derived from our calculation of \( b^2 \), is crucial as it ensures the trinomial is complete.
In the exercise, for \( y^2 + 3y \), we find \( a = y \) and \( b = \frac{3}{2} \). Therefore, our completed binomial square of \((y + \frac{3}{2})^2\) highlights how these two components work seamlessly together.
Quadratic Expressions
Quadratic expressions can often be rearranged to show their underlying structure in the form of binomial squares. A quadratic expression generally takes the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Completing the square allows us to transform it into a perfect square trinomial, revealing its geometric and algebraic properties.
This transformation helps in several ways:
  • Simplifying the equation to identify roots easily.
  • Making further algebraic manipulations more feasible.
By observing that \( y^2 + 3y \) is a simple quadratic missing a constant term, we become aware of its potential to be rewritten as a square of a binomial, namely \( (y + \frac{3}{2})^2 \), once we add the appropriate \( b^2 \) term.

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

Use a graphing utility to graph each equation. $$r^{2}=-2 \sin 2 \theta \text { (lemniscate) }$$

Write each rational number as the quotient of two integers in simplest form. $$0 . \overline{45}$$

Show that \(\left(\begin{array}{l}n \\\ k\end{array}\right)=\left(\begin{array}{c}n \\ n-k\end{array}\right)\) for all positive integers \(n\) and \(k\) with \(0 \leq k \leq n\)

Find the sum of the geometric series. $$\sum_{n=0}^{10} 2(-4)^{n}$$

Use a graphing utility to graph each equation. $$r^{2}=4 \cos 2 \theta \text { (lemniscate) }$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.