Problem 20 - Introductory Algebra for College Students

Question

Solve each quadratic equation using the square root property. Express imaginary solutions in \(a+b i\) form. $$(x+12)^{2}=-100$$

Step-by-Step Solution

Answer

The solutions for the given quadratic equation are \( x = -12 + 10i \) and \( x = -12 - 10i \).

Step 1: Isolate the squared term

To isolate the squared term, keep \( (x+12)^{2} \) untouched and 100 on the other side of the equation, just as given.

Step 2: Apply square root to both sides

The next step is to eliminate the squared term. This can be done by applying a square root to both sides of the equation. Then, the equation will look like \( x+12 = \pm \sqrt{-100} \).

Step 3: Simplify both sides

As the square root of -1 is represented by \(i\), the equation becomes \( x+12 = \pm 10i \).

Step 4: Solve for x

Solve the above equation for \(x\). So subtract 12 from both sides which brings you to \( x = -12 \pm 10i \).

Key Concepts

Square Root Property

The square root property is a crucial concept when solving quadratic equations, especially those in the form \(x^2 = c\). This property allows us to easily find the solutions by taking the square root of both sides of the equation.
This method is particularly helpful when the equation involves a perfect square on one side.

• To use the square root property, ensure that one side of the equation is a perfect square or can be reorganized as one.
• The basic principle is: if \(a^2 = c\), then \(a = \pm \sqrt{c}\).

Once the square root is applied, consider both the positive and negative square roots, because squaring either would result in the same value. In problems like the original exercise where \(x+12)^2 = -100\), applying the square root property gives: \(x+12 = \pm \sqrt{-100}\).
This leads to the need to deal with imaginary numbers, since the square root of a negative number introduces complex numbers.

Imaginary Solutions

Imaginary solutions arise when we take the square root of a negative number. Typically, the square root operation results in real numbers, but when the number under the radical is negative, the solution involves imaginary numbers.
Imaginary numbers are expressed using the symbol \(i\), which is defined as \(i = \sqrt{-1}\).

• For a negative under the square root like \(-100\), this can be split into \(\sqrt{-1} \cdot \sqrt{100}\).
• The \(\sqrt{-1}\) becomes \(i\), therefore \(\sqrt{-100} = 10i\).

So for the step where we see \(x+12=\pm \sqrt{-100}\), recognizing imaginary solutions is key.
Thus, the solutions are \(x+12=\pm 10i\). Imaginary solutions are part of complex solutions, indicating that the quadratic equation has no real intersections with the x-axis.

Complex Numbers

Complex numbers form when we combine real numbers with imaginary numbers. Each complex number has a real part and an imaginary part, expressed in the form \(a+bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
The important thing to note is:

• The real part \(a\) is just like any ordinary real number.
• The imaginary part \(bi\), where \(b\) is also a real number, is multiplied by the imaginary unit \(i\).

Applying this concept to the solutions of the original equation, we get \(x = -12 \pm 10i\).
Here, \(-12\) is the real part and \(\pm 10i\) constitutes the imaginary part.
Complex numbers are incredibly useful across various fields of science and engineering, as they expand the concept of one-dimensional real numbers to a two-dimensional complex plane, making calculations involving square roots of negative numbers, among other things, possible.