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Chapter 8: Problem 125

Explain why there are two square roots for any nonzero real number.

Short Answer

Expert verified
Every nonzero real number has two square roots, with one positive and one negative.

Step by step solution

01

Understanding Square Roots

The square root of a number is a value which, when multiplied by itself, gives the original number. Given a positive number, there are typically two square roots: one positive and one negative. This is because both the positive and negative values, when squared, yield the same positive result.
02

Setting Up the Equation

Consider a nonzero real number, say \( x \). The equation for square roots is \( y^2 = x \), where \( y \) is the square root of \( x \).
03

Solving the Equation

We solve \( y^2 = x \) to find potential values of \( y \). Because the equation \( y^2 = x \) needs to hold, \( y \) can be either the positive or negative root, i.e., \( y = \sqrt{x} \) or \( y = -\sqrt{x} \).
04

Verification

Verify both roots: \( \sqrt{x} \times \sqrt{x} = x \) and \( (-\sqrt{x}) \times (-\sqrt{x}) = x \). Both the positive and negative values satisfy the equation, confirming there are two square roots.

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Key Concepts

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

Real Numbers
Real numbers include all numbers that can be found on the number line. This category encompasses both integers, fractions, and irrational numbers. Real numbers are divided into positive numbers, negative numbers, and zero. Understanding real numbers is crucial to discussing square roots because they describe the values we can take square roots of. In simpler terms:
  • Real numbers can be depicted as points on an infinite line, called the number line.
  • They include rational numbers (which can be expressed as fractions) and irrational numbers (which cannot be precisely written as fractions).
Both positive and negative real numbers have important implications for square roots because they determine the availability of real roots to any equation.
Positive and Negative Roots
Any nonzero real number, when considered in the concept of square roots, typically has two solutions: one positive and one negative. To further illustrate this:
  • When you square a positive number, like 3, the result is positive: 3 x 3 = 9.
  • When you square a negative number, like -3, the result is also positive: (-3) x (-3) = 9.
This symmetry around zero is why both the positive and negative square roots of a real number are valid. For a given real number, both \(+\) and \(-\) roots satisfy the equation \(y^2 = x\), demonstrating this fundamental concept in mathematics. This is true for all non-zero real numbers, making the concept vital in solving equations.
Equation Solving
When solving equations involving square roots, it is essential to recognize that finding a square root is solving the equation \(y^2 = x\). However, the solution might not be straightforward, as there are generally two valid roots.
  • The equation \(y^2 = x\) has two solutions: \(y = \sqrt{x}\) and \(y = -\sqrt{x}\).
  • To solve it, consider the real number \(x\) and determine its square roots.
By applying these solutions, you take into account both the positive and negative square roots, which means both solutions are confirmed by checking that \(\sqrt{x}\ \times \sqrt{x} = x\) and \((-\sqrt{x}) \times (-\sqrt{x}) = x\). This approach ensures you provide every potential solution that satisfies the original equation, reinforcing the duality of square roots.

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