Problem 29 - Precalculus Student Solutions Manual 5th

Question

Solve each equation by the square root property. $$x^{2}=-81$$

Step-by-Step Solution

Answer

The solutions are \(+9i\) and \(-9i\).

Step 1: Set Up the Equation

The given equation is \(x^{2} = -81\). Notice that the left side is a perfect square.

Step 2: Isolate the Square Term

The term \(x^{2}\) is already isolated on one side of the equation, so no further steps are needed to isolate it.

Step 3: Apply the Square Root Property

Take the square root of both sides: \(\sqrt{x^{2}} = \sqrt{-81}\). Remember to include both the positive and negative roots. \(x = \pm \sqrt{-81}\).

Step 4: Simplify the Solution

Recognize that \(-81 = -1 \times 81\) and use the property of square roots to rewrite the expression: \(\pm \sqrt{-1 \times 81} = \pm \sqrt{-1} \times \sqrt{81} = \pm i \times 9 = \pm 9i\). So, the solutions are \(+9i\) and \(-9i\).

Key Concepts

Square Root Property

The square root property is a useful method for solving quadratic equations, especially when the equation is in the form of a perfect square. In general, if we have an equation like \(x^{2} = c\), we can solve for \(x\) by taking the square root of both sides. This gives us two possible solutions: \(x = \pm \sqrt{c}\). The symbol \(\pm \) means 'plus or minus', so we consider both the positive and negative roots.

Applying the square root property to equations involves these steps:

• Ensure one side of the equation has the perfect square term, \(x^{2}\).
• Isolate this term if needed.
• Take the square root of both sides, remembering to include both positive and negative solutions.

When the number inside the square root is negative, it leads us into the realm of imaginary numbers, which we'll cover next.
For example, solving the equation \(x^{2} = -81\), we use the square root property and get:
\(\sqrt{x^{2}} = \sqrt{-81}\)
This will lead to imaginary and complex solutions.

Imaginary Numbers

Imaginary numbers come into play when we need to take the square root of a negative number. A number like \(\sqrt{-1}\) is not possible within the set of real numbers. To address this, mathematicians have defined \(i\) as the imaginary unit, where:
\(\sqrt{-1} = i\)

Key points about imaginary numbers:

• An imaginary number is expressed as a real number multiplied by \(i\).
• The equation \(x^{2} = -81\) from our exercise becomes \(x = \pm 9i\).
• Imaginary numbers allow us to solve equations that have no real solutions.

Understanding imaginary numbers is crucial because they pave the way for complex solutions, which we'll discuss next. Complex solutions are composed of both real and imaginary parts.

Complex Solutions

A complex solution or complex number includes both a real part and an imaginary part. The general form of a complex number is \(a + bi\), where \(a\) represents the real part, and \(bi\) represents the imaginary part.

• In the exercise \(x^{2} = -81\), both solutions are purely imaginary: \(9i\) and \(-9i\).
• Complex solutions are essential in various areas of mathematics and engineering because they provide a means to solve equations beyond the set of real numbers.

When working with complex numbers, it's helpful to understand operations such as addition, subtraction, multiplication, and division. Just remember the property that \(i^{2} = -1\).
Thus, our exercise's final solutions are expressed as complex numbers: \(+9i\) and \(-9i\). This opens a wide array of problem-solving techniques, and ensures our mathematical toolbox is fully equipped.