Problem 84 - Algebra 1: Concepts and Skills

Question

Solve the equation. Write the solutions as integers if possible. Otherwise, write them as radical expressions. (Lesson 9.2) $$ x^{2}-5=20 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 5\) and \(x = -5\).

Step 1: Rearrange the equation

First, we rearrange the equation to isolate \(x^{2}\) on one side. This can be achieved by adding 5 to both sides of the equation: \(x^{2} - 5 + 5 = 20 + 5\). As a result, we have: \(x^{2} = 25\).

Step 2: Square root both sides

Next, we square root both sides of the equation in order to isolate \(x\). As we are dealing with square roots, we must remember there are potentially two solutions, positive and negative. So, we get: \(x = \pm \sqrt{25}\).

Step 3: Solve for variable x

Finally, we simplify the square root to determine our solutions for \(x\). So, our solutions are: \(x = \pm 5\).

Key Concepts

Understanding Radical Expressions

Radical expressions often involve the square root symbol, which is used to denote that a number is being taken to a fractional power. In our exercise, we encountered the square root while solving the equation \(x^2 = 25\), which required finding the value of \(x\). To solve this, we used the square root symbol: \(\sqrt{25}\).

Here's a simple breakdown of this concept:

• The square root symbol (√) is used to denote the principal square root of a non-negative number. For instance, \(\sqrt{25} = 5\).
• For the equation \(x^2 = 25\), finding \(x\) involves taking the square root of both sides.
• A square root equation, such as \(x = \pm \sqrt{25}\), implies there are two solutions: both positive and negative roots.

Radical expressions can be more complex, sometimes requiring simplification if numbers are not perfect squares.

Remember, the square root operation is vital for solving equations involving squared terms, deciphering possible solutions further. By understanding how to work with radical expressions efficiently, like evaluating \(\sqrt{25}\) to determine \(x = \pm 5\), we can streamline solving quadratic equations.

Finding Integer Solutions

Integer solutions are whole numbers that satisfy an equation. In the quadratic equation we worked with, we aimed to express our answer as integers.

Why is this important?

• Integer solutions are simpler to understand and easier to use in further calculations.
• When solving \(x^2 - 5 = 20\), we rearranged the equation to \(x^2 = 25\) and sought the square root.
• This revealed our solutions: \(x = 5\) and \(x = -5\), which are indeed integers.

Not all equations will have tidy, integer solutions. In cases where they don't, radical expressions are the next best choice. However, if an equation's result in square root form simplifies to a whole number, like \(\sqrt{25} = 5\), then you have integer solutions, which offers a cleaner, easier-to-use answer.

Using the Square Root Property

The square root property is an essential technique when solving quadratic equations, especially those representing simple cases like our exercise equation \(x^2 = 25\). By applying this property, we can efficiently discover possible solutions for equations that have a squared term.

The square root property allows us to:

• Take the square root of both sides of an equation, helping us isolate the variable\(x\).
• Remember that a square root can yield two potential answers: positive and negative. Hence, for \(\sqrt{25}\), the possible values for \(x\) are \(\pm 5\).

Understanding this property simplifies solving similar equations by reducing the number of necessary steps. It also helps students recognize patterns, making future problems less intimidating.

When applying the square root property, always check that the equation is simplified, as we did by first converting \(x^2 - 5 = 20\) into \(x^2 = 25\). Knowing the property adds a robust tool to solve specific quadratic equation types.