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Chapter 11: Problem 40

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

Short Answer

Expert verified
x = 卤2鈭�10

Step by step solution

01

- Isolate the Square Term

First, add 80 to both sides of the equation to isolate the term with the square.\[ 2x^2 - 80 + 80 = 0 + 80 \ 2x^2 = 80 \]
02

- Divide to Simplify

Next, divide both sides of the equation by 2 to simplify the equation.\[ \frac{2x^2}{2} = \frac{80}{2} \ x^2 = 40 \]
03

- 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.\[ x = \pm \sqrt{40} \]
04

- 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:\[ x = \pm 2 \sqrt{10} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.
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Most popular questions from this chapter

One solution of \(3 x^{2}-7 x+c=0\) is \(\frac{1}{3} .\) Find \(c\) and the other solution.

Solve each equation. (All solutions are nonreal complex numbers.) $$ x^{2}=-26 $$

Solve for \(x .\) Assume that a and \(b\) represent positive real numbers. \(x^{2}-a^{2}-36=0\)

Solve for \(x .\) Assume that a and \(b\) represent positive real numbers. \((5 x-2 b)^{2}=3 a\)

Solve using the square root property. Simplify all radicals. $$ \left(x+\frac{1}{7}\right)^{2}=\frac{11}{49} $$
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