Solve The Equation By Using The Square-root Property:$\[ 2(x-6)^2 = 8 \\](Type Two Answers As: \[$ X = \#, X = \# \$\])

Introduction

The square-root property is a fundamental concept in algebra that allows us to solve equations involving square roots. In this article, we will focus on solving equations using the square-root property, specifically the equation $2(x-6)^2 = 8$. We will break down the solution step by step and provide a clear explanation of each step.

Understanding the Square-Root Property

The square-root property states that if $x^2 = a$, then $x = \pm \sqrt{a}$. This means that if we have an equation involving a square root, we can eliminate the square root by taking the square of both sides of the equation.

Step 1: Divide Both Sides by 2

To solve the equation $2(x-6)^2 = 8$, we first need to isolate the squared term. We can do this by dividing both sides of the equation by 2.

$$\frac{2(x-6)^2}{2} = \frac{8}{2}$$

This simplifies to:

$$(x-6)^2 = 4$$

Step 2: Take the Square Root of Both Sides

Now that we have isolated the squared term, we can take the square root of both sides of the equation. Remember that when we take the square root of both sides, we must consider both the positive and negative square roots.

$$\sqrt{(x-6)^2} = \sqrt{4}$$

This simplifies to:

$$x-6 = \pm 2$$

Step 3: Solve for x

Now that we have the equation $x-6 = \pm 2$, we can solve for x by adding 6 to both sides of the equation.

$$x = 6 \pm 2$$

This gives us two possible solutions:

$$x = 6 + 2 = 8$$

$$x = 6 - 2 = 4$$

Conclusion

In this article, we solved the equation $2(x-6)^2 = 8$ using the square-root property. We broke down the solution into three steps: dividing both sides by 2, taking the square root of both sides, and solving for x. We found two possible solutions: $x = 8$ and $x = 4$.

Example Problems

Here are a few example problems that you can try to practice solving equations using the square-root property:

• $3(x+2)^2 = 9$
• $2(x-1)^2 = 16$
• $(x+4)^2 = 25$

Tips and Tricks

Here are a few tips and tricks to help you solve equations using the square-root property:

• Make sure to isolate the squared term before taking the square root of both sides.
• Consider both the positive and negative square roots when taking the square root of both sides.
• Be careful when adding or subtracting numbers to both sides of the equation.

Common Mistakes

Here are a few common mistakes to avoid when solving equations using the square-root property:

• Failing to isolate the squared term before taking the square root of both sides.
• Not considering both the positive and negative square roots when taking the square root of both sides.
• Making errors when adding or subtracting numbers to both sides of the equation.

Real-World Applications

The square-root property has many real-world applications in fields such as physics, engineering, and economics. Here are a few examples:

• In physics, the square-root property is used to solve equations involving the motion of objects.
• In engineering, the square-root property is used to solve equations involving the design of structures.
• In economics, the square-root property is used to solve equations involving the behavior of markets.

Conclusion

Introduction

In our previous article, we discussed how to solve equations using the square-root property. In this article, we will answer some frequently asked questions about solving equations using the square-root property.

Q: What is the square-root property?

A: The square-root property is a fundamental concept in algebra that allows us to solve equations involving square roots. It states that if $x^2 = a$, then $x = \pm \sqrt{a}$.

Q: How do I apply the square-root property to solve an equation?

A: To apply the square-root property, you need to follow these steps:

1. Isolate the squared term by dividing both sides of the equation by the coefficient of the squared term.
2. Take the square root of both sides of the equation.
3. Consider both the positive and negative square roots.
4. Solve for x by adding or subtracting the square root to both sides of the equation.

Q: What are some common mistakes to avoid when solving equations using the square-root property?

A: Here are some common mistakes to avoid:

• Failing to isolate the squared term before taking the square root of both sides.
• Not considering both the positive and negative square roots when taking the square root of both sides.
• Making errors when adding or subtracting numbers to both sides of the equation.

Q: Can I use the square-root property to solve equations with negative numbers?

A: Yes, you can use the square-root property to solve equations with negative numbers. However, you need to be careful when taking the square root of both sides of the equation. Remember that the square root of a negative number is an imaginary number.

Q: How do I solve equations with multiple squared terms?

A: To solve equations with multiple squared terms, you need to follow these steps:

1. Isolate one of the squared terms by dividing both sides of the equation by the coefficient of the squared term.
2. Take the square root of both sides of the equation.
3. Consider both the positive and negative square roots.
4. Solve for x by adding or subtracting the square root to both sides of the equation.
5. Repeat the process for the remaining squared terms.

Q: Can I use the square-root property to solve equations with fractions?

A: Yes, you can use the square-root property to solve equations with fractions. However, you need to be careful when taking the square root of both sides of the equation. Remember to simplify the fraction before taking the square root.

Q: How do I check my solutions to make sure they are correct?

A: To check your solutions, you need to plug the solutions back into the original equation and make sure they are true. If the solutions are true, then you have found the correct solutions.

Q: What are some real-world applications of the square-root property?

A: The square-root property has many real-world applications in fields such as physics, engineering, and economics. Here are a few examples:

• In physics, the square-root property is used to solve equations involving the motion of objects.
• In engineering, the square-root property is used to solve equations involving the design of structures.
• In economics, the square-root property is used to solve equations involving the behavior of markets.

Conclusion

In conclusion, the square-root property is a fundamental concept in algebra that allows us to solve equations involving square roots. By following the steps outlined in this article, you can solve equations using the square-root property and apply it to real-world problems. Remember to isolate the squared term, consider both the positive and negative square roots, and be careful when adding or subtracting numbers to both sides of the equation.