/*! 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 20 Solve each equation by using the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 20

Solve each equation by using the Square Root Property. \(x^{2}+8 x+16=7\)

Short Answer

Expert verified
Solutions: \(x = -4 + \sqrt{7}\) and \(x = -4 - \sqrt{7}\).

Step by step solution

01

Set the equation to standard form

Start by writing the equation in the form where one side is a perfect square trinomial. The given equation is \(x^2 + 8x + 16 = 7\). To make one side a perfect square trinomial, we need to subtract 7 from both sides: \(x^2 + 8x + 16 - 7 = 0\). This simplifies to \(x^2 + 8x + 9 = 0\).
02

Recognize the perfect square

Identify the perfect square on the left side of the equation. The expression \(x^2 + 8x + 16\) can be rewritten as \((x + 4)^2\). Thus, our equation becomes \((x + 4)^2 = 7\).
03

Apply the Square Root Property

Apply the Square Root Property to solve \((x + 4)^2 = 7\). This property states that if \(a^2 = b\), then \(a = \pm\sqrt{b}\). Therefore, \(x + 4 = \pm\sqrt{7}\).
04

Solve for x

Isolate \(x\) by subtracting 4 from both sides of the equation: \(x = -4 \pm \sqrt{7}\). This means the solutions are \(x = -4 + \sqrt{7}\) and \(x = -4 - \sqrt{7}\).

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
A perfect square trinomial is a special type of quadratic expression. It can be written in the form \( (a + b)^2 \) or \( (a - b)^2 \). Here we expand these expressions as follows:

  • \( (a + b)^2 = a^2 + 2ab + b^2 \)
  • \( (a - b)^2 = a^2 - 2ab + b^2 \)
In the given problem, our expression \( x^2 + 8x + 16 \) is a perfect square trinomial because it can be rewritten as \( (x + 4)^2 \). Here's why:
  • The term \( x^2 \) suggests the use of \( x \) in our binomial.
  • The term \( 8x \) tells us that \( 2a \times b \) is \( 8 \), therefore \( a = x \) and \( b = 4 \) (since \( 2 \times 4 = 8 \)).
  • The term \( 16 \) is \( b^2 \), confirming \( b = 4 \) (since \( 4 \times 4 = 16 \)).
Recognizing perfect square trinomials helps simplify solving quadratic equations.
Solve Quadratic Equations
Solving quadratic equations involves finding the values of \(x\) that satisfy the equation. In this exercise, we were given the equation \(x^2+8x+16=7\). To solve it, follow these steps:

  • First, rewrite the equation in such a way that it has a perfect square trinomial on one side, as done here to form \((x + 4)^2 = 7\).
  • This simplification helps us easily apply further solving methods like the Square Root Method.
The target is to isolate \(x\) and find its possible values. Using proper techniques for rewriting and solving such equations is essential for finding the correct solutions.
Square Root Method
The Square Root Method is a powerful tool for solving quadratic equations that have been simplified to the form \( (x + a)^2 = b \). Here's how it works in the exercise we examined:

  • Start with the equation \((x + 4)^2 = 7\).
  • Apply the Square Root Property, which tells us if \(a^2 = b\), then \(a = \pm\sqrt{b}\).
  • This process gives us \(x + 4 = \pm\sqrt{7}\).
  • Isolate \(x\) by subtracting 4 from both sides: \(x = -4 \pm \sqrt{7}\).
This results in two solutions for \(x\): \(-4 + \sqrt{7}\) and \(-4 - \sqrt{7}\). This method is efficient and particularly useful when equations are structured to accommodate its application.

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

Write each equation in vertex form. Then identify the vertex, axis of symmetry, and direction of opening. $$ y=\frac{1}{2} x^{2}+6 x+18 $$

Write each equation in vertex form. Then identify the vertex, axis of symmetry, and direction of opening. $$ y=x^{2}-2 x+9 $$

Graph each inequality. $$ y>x^{2}+6 x+5 $$

BUSINESS A mall owner has determined that the relationship between monthly rent charged for store space \(r\) (in dollars per square foot) and monthly profit \(P(r)\) (in thousands of dollars) can be approximated by the function \(P(r)=-8.1 r^{2}+46.9 r-38.2 .\) Solve each quadratic equation or inequality. Explain what each answer tells about the relationship between monthly rent and profit for this mall.

Solve each matrix equation or system of equations by using inverse matrices. $$ \begin{array}{l}{3 j+2 k=8} \\ {j-7 k=18}\end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.