Problem 27 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$(y+8)^{2}=11$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation \((y+8)^{2}=11\) are \(y = \sqrt{11} - 8\) and \(y = - \sqrt{11} - 8\)

Step 1: Apply the square root property

To solve for \(y\) using the square root property, take the square root of both sides of the equation. Remember, the square root of a number, say \(x^{2}\), is \(x\) but in the context of the equation it should be \(\pm x\). Hence, we get: \[y + 8 = \pm \sqrt{11}\]

Step 2: Solve for y

In this step, we want to isolate \(y\) to find its values. Therefore, subtract 8 from both sides of the equation:\[y = \pm \sqrt{11} - 8\]

Step 3: Simplify or rationalize the result

The roots cannot be further simplified since \(11\) is a prime number and \(8\) cannot be divided evenly by \(11\). So, the final solutions are \(y = \sqrt{11} - 8\) and \(y = - \sqrt{11} - 8\)

Key Concepts

Quadratic Equation Solving

Understanding how to solve quadratic equations is essential in algebra. A quadratic equation is typically in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. One effective method to solve a quadratic equation is by using the square root property, which is applied when the equation can be expressed as \( (x + p)^2 = q \). Here's how it works:

First, isolate the squared term on one side of the equation, ensuring that it equals a constant or an expression on the other side. Once you have the squared term by itself, apply the square root to both sides of the equation, which yields \(x + p = \pm\sqrt{q}\), indicating two solutions due to the square root of a number having both positive and negative values. Finally, solve for \(x\) by subtracting or adding any constants. Remember, this property simplifies the process, especially when the quadratic is already in a perfect square form, like in our exercise.

Simplifying Radicals

Simplifying radicals is another continuum of solving quadratic equations, particularly when the solution involves square roots. A radical is in simplest form when the radicand—the number under the radical symbol—has no perfect square factors other than 1, and no fractions or decimals remain.

For example, \(\sqrt{18}\) can be simplified by looking for perfect square factors of 18, which are 9 and 2. Thus, \(\sqrt{18}\) simplifies to \(3\sqrt{2}\) because \(\sqrt{9} \times \sqrt{2} = 3\sqrt{2}\). In the context of our exercise, \(\sqrt{11}\) is already in simplest form because 11 is a prime number and has no perfect square factors. When you cannot simplify the radical further, your answer remains with the radical, as in \(\pm\sqrt{11} - 8\).

Rationalizing Denominators

Rationalizing denominators is a technique used to eliminate radicals from the bottom of a fraction, which makes it easier to understand and work with. A radical in the denominator is not in its simplest form, and in certain fields such as engineering or mathematics, it’s conventional to present an expression without a radical in the denominator.

To rationalize a denominator, you multiply the numerator and the denominator by a radical that will give the denominator a perfect square under the radical sign. For instance, if you have \(1 / \sqrt{3}\), you would multiply the top and bottom by \(\sqrt{3}\) to get \(\sqrt{3} / 3\), a rationalized denominator. This process ensures that all roots are in the numerator and presents the fraction in a more conventional form. While our original exercise doesn't contain a fraction with a radical in the denominator, the rationalizing skill is very useful when faced with such expressions in more complex algebraic operations.