Problem 19 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$(x-3)^{2}=16$$

Step-by-Step Solution

Answer

The solution to the equation are \(x = 7\) and \(x = -1\)

Step 1: Take the Square Root

Alternatively, decide on a positive and negative solution: \(\sqrt{(x-3)^2} = \sqrt{16}\) results in \(x - 3 = 4\) and \(x - 3 = -4\)

Step 2: Solve for x in the Positive Case

Transpose -3 to the other side of the equation from \(x - 3 = 4\) to \(x = 4 + 3\). From the addition of integers, we get \(x=7\)

Step 3: Solve for x in the Negative Case

Do the same for the negative case where \(x - 3 = -4\). When you transpose -3 to the other side, you will get \(x = -4 + 3\). Adding -4 and 3 gives \(x=-1\)

Key Concepts

Square Root Property

The square root property is a method used to solve quadratic equations, especially those in the form \((x-a)^2 = k\). To apply this property, you take the square root of both sides of the equation.
When solving \((x-3)^2 = 16\), you find that:

• \(x - 3 = \sqrt{16}\) (or the positive root)
• \(x - 3 = -\sqrt{16}\) (or the negative root)

These equations provide two solutions because both positive and negative values, when squared, will give the original number \(k\). This method helps you solve equations without needing to expand the squares.

Solving Quadratics

Once the quadratic equation has been transformed using the square root property, solving it becomes straightforward. For example:

• In the positive case, \(x - 3 = 4\). Adding 3 to both sides, you find \(x = 7\).
• In the negative case, \(x - 3 = -4\). By adding 3 to both sides, you get \(x = -1\).

By solving for \(x\) in each scenario, you obtain the two potential solutions: \(x = 7\) and \(x = -1\). This showcases how quadratic equations often have two real solutions.

Simplifying Radicals

Simplifying radicals is a crucial step if the square root involves non-perfect squares. When you have a square root like \(\sqrt{16}\), it's already simplified to 4.
For other numbers, breaking them down into their prime factors helps. For instance:

• \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\)

This process reveals the simplest form of radicals, making calculations easier and cleaner. It’s important to simplify radicals to ensure the solutions are as clear as possible.

Rationalizing Denominators

Sometimes in quadratic solutions, you may end up with a denominator that contains a radical. Rationalizing makes the expression easier to work with by eliminating the radicals.
For example, if your result is \(\frac{1}{\sqrt{2}}\), multiply the numerator and the denominator by \(\sqrt{2}\):

• \(\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)

Rationalizing denominators helps in presenting the solution in a more standard form. This process is often required in mathematical writing to adhere to conventional styles.