Problem 3 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$y^{2}=81$$

Step-by-Step Solution

Answer

The solutions are \(y = 9\) and \(y = -9\).

Step 1: Isolate the squared term

The squared term is already isolated: \(y^2 = 81\)

Step 2: Applying the square root

Taking the square root of both sides to remove the square on \(y\). Remember that the square root could be both positive and negative, thus the equation becomes: \[y = \pm \sqrt{81} \]

Step 3: Calculate and Simplify the square root

Calculate the square root of 81: \[y = \pm 9\] This equation has two solutions, \(y = 9\) or \(y = -9\), the roots are not radicals, so no simplification or rationalization is needed.

Key Concepts

Quadratic Equations

Quadratic equations are an essential component of algebra and form the foundation for various applications in mathematics and the sciences. At their core, quadratic equations are polynomials of the second degree, generally represented in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). The graph of a quadratic equation is a parabola, which can open up or down depending on the sign of \(a\).

The property that characterizes a quadratic equation is the squared term, \(x^2\). This squared element is what necessitates specific methods for solving the equation, such as factoring, completing the square, using the quadratic formula, or employing the square root property, as in the given exercise.

Solving Quadratic Equations

Solving quadratic equations can be approached through various methods, but understanding when and how to apply each method is crucial for efficient problem-solving. When the quadratic equation takes the form \(x^2 = k\), where \(k\) is a positive real number, the square root property becomes an excellent tool. This property states that if \(x^2 = k\), then \(x = \pm\sqrt{k}\).

To apply the square root property, one must isolate the squared term, as was done in Step 1 of the example solution. After isolating, taking the square root of both sides of the equation, as in Step 2, allows us to consider both the positive and negative roots. This essential step acknowledges the principle that a squared number can originate from either a positive or a negative number. Then, the student calculates and simplifies the square root to obtain the solutions.

Simplifying Radicals

When solving quadratic equations, you may often encounter square roots of numbers. Simplifying radicals refers to the process of finding the simplest radical term that is equivalent to the given radical expression. When the radicand (the number under the radical symbol) is a perfect square, the simplification results in an integer, as was the case with \(\sqrt{81}\) which simplified to 9.

If the radicand is not a perfect square, simplifying the radical involves factoring the radicand into its prime factors and pairing any identical factors to move them outside the radical symbol. If the solution includes an irrational number, it is sometimes necessary to rationalize the denominator. Rationalizing involves removing the radical from the denominator of a fraction, which was not required in our example as the roots were integers. It's important to verify if the radical can be simplified to ensure the most straightforward form of the answer.