Problem 40 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ 2 x^{2}-80=0 $$

Step-by-Step Solution

Answer

x = ±2√10

Step 1 - Isolate the Square Term

First, add 80 to both sides of the equation to isolate the term with the square.\[ 2x^2 - 80 + 80 = 0 + 80 \ 2x^2 = 80 \]

Step 2 - Divide to Simplify

Next, divide both sides of the equation by 2 to simplify the equation.\[ \frac{2x^2}{2} = \frac{80}{2} \ x^2 = 40 \]

Step 3 - Apply the Square Root Property

Now, apply the square root property to both sides of the equation. Remember to include both the positive and negative roots.\[ x = \pm \sqrt{40} \]

Step 4 - Simplify the Radical

Finally, simplify the radical expression by breaking it down into its prime factors.\[ \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2 \sqrt{10} \]Thus, the solutions are:\[ x = \pm 2 \sqrt{10} \]

Key Concepts

Square Root Property

The square root property is a useful tool when dealing with quadratic equations. Let's understand it better. When you have an equation in the form of \(x^2 = a\), you can find the solutions by taking the square root of both sides.
Remember, the square root of a number has two values: the positive root and the negative root. Hence, the solution to \(x^2 = a\) is \(x = \pm \sqrt{a}\).
This property simplifies the problem significantly, making it easier to isolate the variable and solve the equation.

Simplifying Radicals

When we apply the square root property, we often end up with roots that can be simplified. For instance, in the equation \(2x^2 = 80\), solving it gives the intermediate step \(x = \sqrt{40}\).
To simplify \sqrt{40}\, break it down into its prime factors.

• \