Problem 6 - Algebra A Combined Function

Question

Use the square root property to solve each equation. $$ y^{2}=20 $$

Step-by-Step Solution

Answer

The solutions are \(y = \pm 2\sqrt{5}\).

Step 1: Identify the Equation Format

The given equation is in the form of a perfect square: \(y^2 = 20\). To solve it, we will use the square root property.

Step 2: Apply the Square Root Property

Using the square root property, which states that if \(x^2 = a\), then \(x = \pm \sqrt{a}\), apply it to the equation: \(y = \pm \sqrt{20}\).

Step 3: Simplify the Square Root

The square root of 20 can be simplified by finding its prime factors. Since \(20 = 4 \times 5\) and \(\sqrt{4} = 2\), we can write \(\sqrt{20}\) as \(2\sqrt{5}\).

Step 4: Write the Final Solution

By substituting back the simplified form, we have the solutions: \(y = 2\sqrt{5}\) and \(y = -2\sqrt{5}\).

Key Concepts

Solving Quadratic Equations

Quadratic equations are a type of polynomial equation characterized by the presence of a squared term. These equations generally take the form:

• ax² + bx + c = 0

In many cases, such as our original exercise where the equation is simplified to y² = 20, we are only dealing with one term squared. For these, we can apply simpler methods such as the square root property to find our solution.

The square root property is effective because it allows us to solve equations of the form x² = a directly. Instead of moving terms around and factoring, we can jump to finding the value of 'x' by taking the square root of both sides of the equation. Remember that taking the square root of a number will yield two possible solutions: a positive and a negative value, denoted as plus-minus (±). Thus, for y² = 20, the solutions are y = ±√20.

Simplifying Square Roots

Square roots often appear in equations, and simplifying them is crucial for obtaining neat and manageable results. When simplifying square roots, the goal is to express the square root in its most simplified form by finding factors that are perfect squares.

Take the number 20 as an example. Its prime factorization is 2 × 2 × 5, or more intuitively, 4 × 5. The number 4 is a perfect square because

• 4 = 2²

Thus, the square root of 20 becomes:

• √20 = √(4 × 5) = √4 × √5 = 2√5.

The process of simplifying square roots through factorization is essential in many mathematical problems and allows for a cleaner, more concise representation of the solution.

Perfect Square Equations

A perfect square equation is an equation that can be expressed as the square of a binomial. It typically appears in scenarios where a term is squared, making it possible to use the square root property quickly.

Consider an equation like y² = 20 from our exercise. Though '20' itself is not a perfect square, the left-hand side of the equation, y², suggests that it once came from squaring a binomial—here, simply a single variable 'y'.

Perfect squares, when processed correctly, simplify the solving process by allowing direct application of properties and result in clearer solutions. Identifying and working with perfect square equations typically follow recognizing equations of the form y² = a and then utilizing tools like square root property to resolve them without jumping into more complex algebraic manipulations.

Radical Expressions

Radical expressions often include square roots and require careful manipulation to simplify or solve equations. A radical expression can include variables under the radical, roots other than square roots, or coefficients.

When working with radical expressions, especially involving integers and perfect square factors, the primary goal is simplification. For instance, in the equation y = ±√20, the first step is to factor 20 into its simplest radical components, resulting in 2√5.

Such manipulation helps in not only simplifying but also in understanding the expression better, especially during multiplication or addition of radical expressions. Remember to follow algebraic rules when dealing with these expressions to ensure the accuracy and correctness of your mathematical procedures.