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Chapter 7: Problem 16

Find all of its square roots. $$ 225 $$

Short Answer

Expert verified
The square roots of 225 are 15 and -15.

Step by step solution

01

Understand the Problem

Identify what is being asked: Find the square roots of 225. Remember, a square root of a number is a value that, when multiplied by itself, gives the original number.
02

Express in Equation Form

Write the equation to find the square roots of 225. We can express this as \[ x^2 = 225 \]
03

Solve the Equation

To find the square roots, take the square root of both sides of the equation: \[ x = \pm \sqrt{225} \]
04

Calculate the Square Root

Calculate the principal square root of 225: \[ \sqrt{225} = 15 \]. Remember, there are two solutions: one positive and one negative.
05

State the Solutions

State the square roots of 225: \[ x = 15 \] and \[ x = -15 \].

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Key Concepts

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

Solving Equations
When you need to find the square root of a number, you often start by writing an equation. Here, we want to find the square roots of 225. We can write this as an equation: \[ x^2 = 225 \]This equation means we are looking for a number \(x\) that, when multiplied by itself (or squared), equals 225. Solving equations like this is a fundamental skill in algebra. Once you have the equation set up, you can start solving for \(x\). Remember, solving equations usually involves isolating the variable (in this case, \(x\)) to find its value.
Square Root
The square root of a number is a value that, when multiplied by itself, gives the original number. For the number 225, we need to find its square roots. We write the equation: \[ x = \pm \sqrt{225} \]Taking the square root of both sides of the equation gives us two values: one positive and one negative. The principal (or positive) square root of 225 is 15. Therefore, \sqrt{225} = 15\. The square root function has this property of giving both positive and negative values because both \(15^2 = 225\) and \((-15)^2 = 225\).
Positive and Negative Solutions
Square roots can have both positive and negative solutions. This is because squaring a positive number or its negative counterpart results in the same positive number. For instance:
  • 15 multiplied by itself (15 \times 15) gives 225.
  • Similarly, -15 multiplied by itself (-15 \times -15) also gives 225.
Thus, the equation has two solutions: \[ x = 15 \] and \[ x = -15 \].Understanding that there are always two roots (one positive and one negative) for square roots of a positive number is key. When dealing with real numbers, you will usually end up with two solutions for problems involving square roots.

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Most popular questions from this chapter

Speed of a Skidding Car. Police can estimate the speed at which a car was traveling by measuring its skid marks. The function given by $$r(L)=2 \sqrt{5 L}$$ can be used, where \(L\) is the length of a skid mark, in feet, and \(r(L)\) is the speed, in miles per hour. Find the exact speed and an estimate (to the nearest tenth mile per hour) for the speed of a car that left skid marks (a) 20 ft long; (b) 70 ft long; (c) 90 ft long. See also Exercise \(102 .\)

Multiply. Assume that all variables represent nonnegative real numbers. $$ \sqrt{3}(6-\sqrt{3}) $$

Add or subtract. Simplify by combining like radical terms, if possible. Assume that all variables and radicands represent positive real numbers. $$ \sqrt{4 x-4}-\sqrt{x^{3}-x^{2}} $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ (r-\sqrt[4]{s^{3}})(3 r-\sqrt[5]{s}) $$

Simplify. $$ \frac{1}{4+\sqrt{3}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{3}-4} $$
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