Problem 4 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$y^{2}=144$$

Step-by-Step Solution

Answer

The solution to the equation \(y^{2}=144\) is \(y = 12, -12\).

Step 1: Understand the Problem

The equation is \(y^2 = 144\). According to the square root property, if \(x^2 = c\) then \(x = \pm \sqrt{c}\). So to find 'y', the square root of 144 must be taken which yields two results, the positive and negative square root.

Step 2: Taking square root of both sides

Take the square root of both sides of the equation. This gives \(\sqrt{y^2} = \pm \sqrt{144}\), which simplifies to \(y = \pm 12\) because the square root of 144 is 12.

Key Concepts

Square Root Property

The square root property is a valuable tool in solving quadratic equations of the form \(x^2 = c\). This property states that if \(x^2 = c\), then \(x = \pm \sqrt{c}\).
This means you should consider both the positive and negative square roots of \(c\). For instance, in the equation \(y^2 = 144\), applying the square root property involves finding both \(+\sqrt{144}\) and \(-\sqrt{144}\), which result in \(y = 12\) and \(y = -12\).

By using the square root property, you can quickly find solutions to quadratic equations where the variable is squared. It's a method that bypasses some of the more complex processes like factoring or completing the square when the equation is set to zero. Remember, not every quadratic equation has integer solutions, but when it does, the square root property often highlights them in a straightforward manner.
The key takeaway is: when faced with \(x^2 = c\), leverage the power of the square root property to simplify to \(x = \pm \sqrt{c}\), ensuring you account for both possible solutions.

Radicals

Radicals, often represented by the square root symbol \(\sqrt{}\), are mathematical expressions that indicate the root of a number or expression. A common radical is the square root, where you're trying to find a number that, when multiplied by itself, yields the radicand (the number inside the root).

For example, in \(\sqrt{144}\), 144 is the radicand. Simplifying a radical like this involves finding the square root, which is 12 in this case because \(12 \times 12 = 144\). Simplifying radicals is a crucial process when dealing with equations that result in non-integer radicands. Simplified radicals make equations easier to handle and understand.

Radicals can sometimes involve more than just simple square roots. Higher order roots like cube roots (\(\sqrt[3]{x}\)) and fourth roots (\(\sqrt[4]{x}\)) also exist, but the same principles apply. Identifying and simplifying radicals helps maintain accuracy and simplicity in solving mathematical problems. Remember, always attempt to express radicals in their simplest form possible.

Rationalizing Denominators

Rationalizing the denominator is a process used to eliminate radicals from the bottom of a fraction, making it easier to deal with. This step is often necessary in mathematical operations because it simplifies the expression's handling and clarifies the relationship between numbers.

To rationalize a denominator, multiply the numerator and the denominator by the radical present in the denominator. As an example, if you began with \(\frac{1}{\sqrt{2}}\), you would multiply the top and bottom of the fraction by \(\sqrt{2}\), yielding \(\frac{\sqrt{2}}{2}\). Now the denominator is rational (without a radical), and the fraction is more straightforward to use.

The goal is always to have a rational number in the denominator. This makes it easier to add, subtract, and compare fractions.

• Identify the radical in the denominator.
• Multiply the numerator and denominator by this radical.
• Simplify the resulting fraction.

Rationalizing denominators maintains mathematical precision and allows clear, concise computations in all types of algebraic processes.