Problem 30 - Precalculus Student Solutions Manual 5th

Question

Solve each equation by the square root property. $$x^{2}=-400$$

Step-by-Step Solution

Answer

The solutions are \(\text{x = }\text{±}20i\).

Step 1: Recognize the Structure

Identify that the equation is in the form of a quadratic equation where the variable is squared: \(x^2 = -400\).

Step 2: Understand the Square Root Property

The square root property states that if \(x^2 = c\), then \(x = \pm \sqrt{c}\). However, this only applies to non-negative values of \(c\). If \(c = -400\), consider the square root of a negative number.

Step 3: Address the Negative Under the Square Root

Recognize that \sqrt{-1} = i\), where \(i\) is the imaginary unit. Thus, \(\text{for} \ c = -400 \, apply the imaginary unit: \(x^2 = -400\) => \(x = \pm \sqrt{-400} = \pm \sqrt{400} \cdot \sqrt{-1} = \pm 20i\).

Key Concepts

square root property

The square root property is a fundamental tool for solving quadratic equations. This property states that if you have an equation in the form of: \[ x^2 = c \], where \(c\) is a constant, you can solve for \(x\) by taking the square root of both sides. This gives you two solutions: \(x = \pm \sqrt{c}\). However, this rule only applies when \(c\) is a non-negative value.

For example, if \(x^2 = 9\), then you can solve it by taking the square root: \(x = \pm \sqrt{9}\), which simplifies to \(x = \pm 3\).

When the constant \(c\) is negative, the process requires an additional step involving imaginary numbers.

imaginary numbers

Imaginary numbers are used to handle the square root of negative numbers, which is not possible within the real number system. The imaginary unit is denoted as \(i\), where: \[ i = \sqrt{-1} \].

This means \(i^2 = -1\). Using this definition, you can rewrite the square root of any negative number. For instance, \(\sqrt{-400}\) can be expressed as: \[ \sqrt{-400} = \sqrt{400 \cdot -1} = \sqrt{400} \cdot \sqrt{-1} = 20i \].

So when you come across equations like \(x^2 = -400\), you must incorporate \(i\) to represent the imaginary component of the solution.

complex solutions

Complex solutions are solutions that include both a real part and an imaginary part. When solving quadratic equations with negative constants, your solutions will typically be in the form of complex numbers.

For the quadratic equation \(x^2 = -400\), the solutions are not real numbers, but rather complex numbers. By applying the square root property and incorporating imaginary numbers, the solutions are found as follows: \[ x = \pm \sqrt{-400} = \pm 20i \].

These solutions are purely imaginary since they have no real number part. When expressed in standard form, they appear as \(0 \pm 20i\). This example illustrates the process of arriving at complex solutions using the square root property and imaginary numbers.