Solve Each Equation By Taking Square Roots

Solving equations by taking square roots is a fundamental technique in algebra. It's a powerful and efficient method, especially when dealing with equations where the variable is squared. This method allows us to isolate the variable and find its possible values by understanding the properties of square roots and their inverse relationship with squaring. Let's delve into the process of solving equations by taking square roots, explore the underlying principles, and examine various examples to solidify your understanding.

Understanding the Basics

The foundation of solving equations by taking square roots lies in the inverse relationship between squaring a number and taking its square root. In simple terms, if we square a number, say x, to get x², then taking the square root of x² will give us back the original number, x. However, it’s crucial to remember that any positive number has two square roots: a positive square root and a negative square root. For instance, the square root of 9 is both 3 and -3, because 3² = 9 and (-3)² = 9.

Key Concepts:

• Square Root: A value that, when multiplied by itself, equals a given number.
• Inverse Operation: An operation that undoes another operation. Squaring and taking the square root are inverse operations.
• Positive and Negative Roots: Every positive number has two square roots: one positive and one negative. This is because squaring either a positive or a negative number results in a positive number.
• Radical Symbol: The symbol √, which denotes the square root operation.

Steps to Solve Equations by Taking Square Roots

The process of solving equations by taking square roots typically involves the following steps. Let's break them down:

Step 1: Isolate the Squared Term

The first and most important step is to isolate the term that is being squared on one side of the equation. This means performing algebraic operations (addition, subtraction, multiplication, division) to get the squared term by itself.

Example:

Consider the equation: 3x² - 5 = 10

To isolate the x² term, we need to:

1. Add 5 to both sides: 3x² = 15
2. Divide both sides by 3: x² = 5

Now, the squared term (x²) is isolated on one side of the equation.

Step 2: Take the Square Root of Both Sides

Once the squared term is isolated, take the square root of both sides of the equation. Remember to consider both the positive and negative square roots.

Example (Continuing from above):

We have x² = 5.

Taking the square root of both sides gives us:

√(x²) = ±√5

This simplifies to:

x = ±√5

Step 3: Simplify the Square Root (if possible)

After taking the square root, simplify the resulting expression if possible. This might involve simplifying radicals or combining like terms.

Example (Continuing from above):

In our example, √5 cannot be simplified further because 5 is a prime number. Therefore, the solutions are:

x = √5 and x = -√5

Step 4: Write Down Both Solutions

As mentioned earlier, always remember that a positive number has two square roots. Write down both the positive and negative solutions to the equation.

Example (Final Solutions):

The solutions to the equation 3x² - 5 = 10 are:

x = √5

x = -√5

These can be written more compactly as x = ±√5.

Examples with Detailed Explanations

Let's work through several examples to illustrate the process and address different scenarios:

Example 1: Simple Equation

Solve for x: x² = 16

1. Isolate the squared term: The x² term is already isolated.
2. Take the square root of both sides: √(x²) = ±√16
3. Simplify: x = ±4
4. Write down both solutions: x = 4, x = -4

Therefore, the solutions are x = 4 and x = -4.

Example 2: Equation with a Coefficient

Solve for x: 2x² = 50

1. Isolate the squared term: Divide both sides by 2: x² = 25
2. Take the square root of both sides: √(x²) = ±√25
3. Simplify: x = ±5
4. Write down both solutions: x = 5, x = -5

Therefore, the solutions are x = 5 and x = -5.

Example 3: Equation with Addition/Subtraction

Solve for x: x² + 3 = 12

1. Isolate the squared term: Subtract 3 from both sides: x² = 9
2. Take the square root of both sides: √(x²) = ±√9
3. Simplify: x = ±3
4. Write down both solutions: x = 3, x = -3

Therefore, the solutions are x = 3 and x = -3.

Example 4: Equation with Parentheses

Solve for x: ( x - 2 )² = 25

1. Isolate the squared term: The ( x - 2 )² term is already isolated.
2. Take the square root of both sides: √(( x - 2 )²) = ±√25
3. Simplify: x - 2 = ±5
4. Solve for x: Add 2 to both sides: x = 2 ± 5
5. Write down both solutions:
   • x = 2 + 5 = 7
   • x = 2 - 5 = -3

Therefore, the solutions are x = 7 and x = -3.

Example 5: Equation with a Fraction

Solve for x: 4x² - 1 = 0

1. Isolate the squared term:
   • Add 1 to both sides: 4x² = 1
   • Divide both sides by 4: x² = 1/4
2. Take the square root of both sides: √(x²) = ±√(1/4)
3. Simplify: x = ±1/2
4. Write down both solutions: x = 1/2, x = -1/2

Therefore, the solutions are x = 1/2 and x = -1/2.

Example 6: Dealing with No Real Solutions

Solve for x: x² + 9 = 0

1. Isolate the squared term: Subtract 9 from both sides: x² = -9
2. Take the square root of both sides: √(x²) = ±√(-9)

Here, we encounter a problem. The square root of a negative number is not a real number. Therefore, this equation has no real solutions. The solutions are imaginary numbers (involving the imaginary unit i, where i² = -1), but we're focusing on real number solutions in this context.

When to Use this Method

Solving by taking square roots is most effective when:

• The equation can be easily manipulated to isolate a squared term.
• The equation does not contain a linear term (i.e., a term with x to the power of 1). If there's a linear term, other methods like factoring, completing the square, or using the quadratic formula are more appropriate.
• You're looking for real number solutions. If the equation leads to taking the square root of a negative number, there are no real solutions.

Common Mistakes to Avoid

• Forgetting the Negative Root: This is the most common mistake. Always remember to consider both the positive and negative square roots.
• Not Isolating the Squared Term: You must isolate the squared term before taking the square root.
• Incorrect Simplification: Make sure to simplify the square root correctly.
• Ignoring No Real Solutions: Be aware that some equations will have no real solutions if taking the square root leads to a negative number.

Advanced Examples and Applications

Let's explore some more complex examples and see how this method can be applied in different contexts.

Example 7: Equation with Multiple Terms

Solve for x: 5( x + 1 )² - 20 = 0

1. Isolate the squared term:
   • Add 20 to both sides: 5( x + 1 )² = 20
   • Divide both sides by 5: ( x + 1 )² = 4
2. Take the square root of both sides: √(( x + 1 )²) = ±√4
3. Simplify: x + 1 = ±2
4. Solve for x: Subtract 1 from both sides: x = -1 ± 2
5. Write down both solutions:
   • x = -1 + 2 = 1
   • x = -1 - 2 = -3

Therefore, the solutions are x = 1 and x = -3.

Example 8: Application in Geometry

Suppose you have a square with an area of 64 square units. Find the length of one side of the square.

• Let s be the length of one side of the square.
• The area of the square is given by s² = 64.
• To find s, take the square root of both sides: √(s²) = ±√64
• Simplify: s = ±8

Since the length of a side cannot be negative, we take the positive solution: s = 8.

Therefore, the length of one side of the square is 8 units.

Example 9: Combining Like Terms

Solve for x: 3x² + 5 - x² = 17

1. Combine like terms: 2x² + 5 = 17
2. Isolate the squared term:
   • Subtract 5 from both sides: 2x² = 12
   • Divide both sides by 2: x² = 6
3. Take the square root of both sides: √(x²) = ±√6
4. Simplify: x = ±√6
5. Write down both solutions: x = √6, x = -√6

Therefore, the solutions are x = √6 and x = -√6.

Theoretical Underpinnings

The method of solving equations by taking square roots relies on the following mathematical principles:

• Definition of Square Root: The square root of a number a is a number b such that b² = a.
• Property of Equality: If a = b, then √a = ±√b. This property allows us to perform the same operation (taking the square root) on both sides of the equation without changing the equality.
• Inverse Relationship: Squaring and taking the square root are inverse operations, meaning they undo each other.

Understanding these principles provides a solid foundation for applying this method correctly and confidently.

Practice Problems

To reinforce your understanding, try solving the following equations:

1. x² = 81
2. 4x² = 36
3. x² - 7 = 2
4. ( x + 3 )² = 16
5. 2( x - 1 )² = 8
6. 9x² - 4 = 0
7. x² + 5 = 1

Conclusion

Solving equations by taking square roots is a valuable technique in algebra that simplifies the process of finding solutions when dealing with squared variables. By isolating the squared term, taking the square root of both sides (remembering both positive and negative roots), and simplifying, you can efficiently solve a variety of equations. Mastering this method requires a clear understanding of the properties of square roots, the inverse relationship between squaring and taking the square root, and careful attention to detail. Remember to watch out for common mistakes, such as forgetting the negative root or failing to isolate the squared term properly. With practice, you'll become proficient in using this technique to solve equations and tackle more complex algebraic problems.