Problem 29 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$(z-4)^{2}=18$$

Step-by-Step Solution

Answer

The solutions to the equation \((z-4)^{2}=18\) are \(z=4+3\sqrt{2}\) and \(z=4-3\sqrt{2}\).

Step 1: Apply the Square Root Property

The square root property is given by: If \(x^{2} = k\), then \(x=\sqrt{k}\) or \(x=-\sqrt{k}\). For the given quadratic equation \((z-4)^{2}=18\), take the square root of both sides to give \(z-4=\sqrt{18}\) and \(z-4=-\sqrt{18}\). Remember to consider the positive and negative values when taking square roots in this context.

Step 2: Simplify the Roots

The root 18 can be simplified as \(\sqrt{9*2} = 3\sqrt{2}\). Substitute this into the equations to give us \(z-4=3\sqrt{2}\) and \(z-4=-3\sqrt{2}\).

Step 3: Solve for z

Now solve the above two equations for z. The solutions to the equation are \(z=4+3\sqrt{2}\) and \(z=4-3\sqrt{2}\).

Key Concepts

Understanding the Square Root Property

The square root property is a useful tool for solving quadratic equations that can be expressed in the form \((x-a)^2=k\). It allows us to directly solve for the variable by taking the square root of both sides of the equation. This property tells us that if \(x^2 = k\), then there are two potential solutions: \(x = \sqrt{k}\) or \(x = -\sqrt{k}\). Remember, whenever you apply the square root, you must consider both the positive and negative roots. Having both positive and negative roots ensures all possible solutions are accounted for.

For instance, in our example \((z-4)^2 = 18\), applying the square root property yields two equations:

• \(z-4 = \sqrt{18}\)
• \(z-4 = -\sqrt{18}\)

This step is crucial because it simplifies our problem from a quadratic equation to two simpler linear equations, which are much more straightforward to solve. By understanding this property, you'll find solving certain types of quadratic equations becomes less daunting.

Tips for Simplifying Radicals

Simplifying radicals involves breaking down a square root into its simplest form. This process is essential in making your final answer more manageable and easier to interpret. The first step is identifying any perfect square factors of the number under the square root.

For example, with \(\sqrt{18}\), we notice that 18 can be factored into \(9 \times 2\). Since 9 is a perfect square, we can simplify \(\sqrt{18}\) to \(\sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}\). Breaking down the radical this way simplifies the expression and makes calculations cleaner.

When simplifying radicals, keep in mind:

• Look for the largest perfect square factor.
• Express it as the product of perfect squares and other integers.
• Simplify separately and multiply the results.

Simplifying radicals not only helps in getting your solution in its simplest form but also avoids unnecessarily complex expressions when communicating your final answer.

Mastering Algebraic Equation Solving

Algebraic equation solving is a key skill in mathematics. It involves finding the value of the variable that makes an equation true. When solving algebraic equations derived from quadratic expressions using the square root property, it's important to consolidate the results by finding each value of the variable.

In our original problem, after applying the square root property and simplifying the radicals, we have two linear equations:

• \(z-4 = 3\sqrt{2}\)
• \(z-4 = -3\sqrt{2}\)

To solve these, we isolate \(z\) by adding 4 to both sides of each equation:

• \(z = 4 + 3\sqrt{2}\)
• \(z = 4 - 3\sqrt{2}\)

Each solution is derived from addressing each equation individually. This process might look simple, but it emphasizes how algebra allows us to approach problems systematically and logically. It's a step-by-step approach, ensuring that each action taken is justified and leads you closer to understanding and solving any quadratic equation effectively.