/*! 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 Find all real solutions to each ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 20

Find all real solutions to each equation. $$(a-2)^{2}=25$$

Short Answer

Expert verified
The solutions are \( a = 7 \) and \( a = -3 \).

Step by step solution

01

Understand the equation

We start with the equation \( (a-2)^2 = 25 \). This is a quadratic equation in terms of \( a-2 \). To solve it, we'll need to take the square root of both sides.
02

Take the square root

Taking the square root of both sides of the equation, we get: \[(a-2) = \pm \sqrt{25}\]. Since \( \sqrt{25} = 5 \), this simplifies to: \[(a-2) = \pm 5\].
03

Solve for \( a \)

Now we solve for \( a \) in both cases.**Case 1**: \( a-2 = 5 \)Adding 2 to both sides gives \( a = 7 \).**Case 2**: \( a-2 = -5 \)Adding 2 to both sides gives \( a = -3 \).
04

Verify the solutions

Substitute both solutions back into the original equation to verify their correctness.For \( a = 7 \): \((7-2)^2 = 5^2 = 25 \).For \( a = -3 \): \((-3-2)^2 = (-5)^2 = 25 \).Both solutions satisfy the original equation.

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.

square root method
The square root method is a common technique to solve quadratic equations in the form \( (x - h)^2 = k \). This form indicates that we're dealing with a squared term.
Here's how it works:

When you face such an equation, you first isolate the squared term.
Then, you take the square root of both sides of the equation.
Remember: taking the square root introduces two solutions: a positive and a negative one.

For example:
Given: \ (a-2)^2 = 25 \
Step 1: Take the square root of both sides: \a-2 = \pm\sqrt{25} \.
Step 2: Since \sqrt{25} = 5\, this becomes \ a-2 =\textbackslashpm 5 \.

This method is straightforward but powerful, especially when identifying both the positive and negative roots.
verification of solutions
Verification of solutions is a crucial step after solving an equation. It ensures that the solutions obtained are correct and satisfy the original equation.

Here's how you verify solutions:
  • Substitute each solution back into the original equation.
  • Check if both sides of the equation are equal.

Example from our exercise:
After solving \( (a-2)^2 = 25 \), we got \a=7\ and \a=-3\.
To verify:
* For \a=7\: Substitute into the original equation:\ (7-2)^2 = 5^2 = 25\.
* For \a=-3\: Substitute into the original equation:\ (-3-2)^2 = (-5)^2 = 25\.

You see, both solutions satisfy the original equation, thus verifying their correctness.
properties of quadratics
Quadratic equations have unique properties that make them identifiable and solvable. A quadratic equation is generally given in the standard form \( ax^2 + bx + c = 0 \). Here are key properties and features:
  • Parabolic Shape: Quadratic equations graph as parabolas, which can either open upwards or downwards.
  • Vertex Form: Parabolas have a vertex which is the maximum or minimum point, depending on the direction the parabola opens.
  • Roots/Solutions: These are the x-values where the quadratic crosses the x-axis (solutions to \( ax^2 + bx + c = 0 \)).
  • Symmetry: Parabolas are symmetric about a vertical line through the vertex.


In our example:

The equation \( (a-2)^2 = 25 \) shows the shape of \( (x-h)^2 \).
Identifying the vertex and solving for the roots helps us understand the full solution landscape and symmetry of the problem.

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

Solve each equation. $$\sqrt[3]{2-w}=\sqrt[3]{2 w-28}$$

Work in a small group to write a formula that gives the side of a cube in terms of the volume of the cube and explain the formula to the other groups.

Simplify. $$\frac{\sqrt{6}}{2} \cdot \frac{1}{\sqrt{3}}$$

Solve each problem by writing an equation and solving it. Find the exact answer and simplify it using the rules for radicals. Find the length of the side of a square sign whose area is 50 square feet.

Simplify. $$\frac{\sqrt{6}}{\sqrt{7}} \cdot \frac{\sqrt{14}}{\sqrt{3}}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Pure 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.