Solve For \[$ X \$\] In The Equation:$\[ \sqrt{x+7} - 2 = X - 1 \\]

by ADMIN 68 views

Introduction

In this article, we will delve into solving for x in the given equation: x+7βˆ’2=xβˆ’1\sqrt{x+7} - 2 = x - 1. This equation involves a square root and a linear term, making it a bit more complex than a standard linear equation. We will use algebraic manipulations to isolate the variable x and find its value.

Step 1: Isolate the Square Root Term

The first step is to isolate the square root term on one side of the equation. We can do this by adding 2 to both sides of the equation:

x+7βˆ’2+2=xβˆ’1+2\sqrt{x+7} - 2 + 2 = x - 1 + 2

This simplifies to:

x+7=x+1\sqrt{x+7} = x + 1

Step 2: Square Both Sides

To eliminate the square root, we can square both sides of the equation:

(x+7)2=(x+1)2(\sqrt{x+7})^2 = (x + 1)^2

This simplifies to:

x+7=x2+2x+1x + 7 = x^2 + 2x + 1

Step 3: Rearrange the Equation

We can rearrange the equation to get all the terms on one side:

x2+2x+1βˆ’xβˆ’7=0x^2 + 2x + 1 - x - 7 = 0

This simplifies to:

x2+xβˆ’6=0x^2 + x - 6 = 0

Step 4: Factor the Quadratic Equation

The quadratic equation x2+xβˆ’6=0x^2 + x - 6 = 0 can be factored as:

(x+3)(xβˆ’2)=0(x + 3)(x - 2) = 0

Step 5: Solve for x

To solve for x, we can set each factor equal to zero and solve for x:

x+3=0β‡’x=βˆ’3x + 3 = 0 \Rightarrow x = -3

xβˆ’2=0β‡’x=2x - 2 = 0 \Rightarrow x = 2

Conclusion

In this article, we solved for x in the equation x+7βˆ’2=xβˆ’1\sqrt{x+7} - 2 = x - 1. We used algebraic manipulations to isolate the variable x and find its value. The final solution is x = -3 or x = 2.

Checking the Solutions

To check the solutions, we can plug them back into the original equation:

x+7βˆ’2=xβˆ’1\sqrt{x+7} - 2 = x - 1

For x = -3:

βˆ’3+7βˆ’2=βˆ’3βˆ’1\sqrt{-3+7} - 2 = -3 - 1

4βˆ’2=βˆ’4\sqrt{4} - 2 = -4

2βˆ’2=βˆ’42 - 2 = -4

This is true, so x = -3 is a valid solution.

For x = 2:

2+7βˆ’2=2βˆ’1\sqrt{2+7} - 2 = 2 - 1

9βˆ’2=1\sqrt{9} - 2 = 1

3βˆ’2=13 - 2 = 1

This is true, so x = 2 is also a valid solution.

Final Answer

Introduction

In our previous article, we solved for x in the equation x+7βˆ’2=xβˆ’1\sqrt{x+7} - 2 = x - 1. We used algebraic manipulations to isolate the variable x and find its value. In this article, we will answer some frequently asked questions related to the solution.

Q: What is the final answer to the equation?

A: The final answer to the equation is x = -3 or x = 2.

Q: How did you isolate the square root term?

A: We isolated the square root term by adding 2 to both sides of the equation. This allowed us to get the square root term on one side of the equation.

Q: Why did you square both sides of the equation?

A: We squared both sides of the equation to eliminate the square root. This allowed us to get rid of the square root and work with a simpler equation.

Q: How did you factor the quadratic equation?

A: We factored the quadratic equation by finding two numbers whose product is -6 and whose sum is 1. These numbers are -3 and 2, so we can write the quadratic equation as (x + 3)(x - 2) = 0.

Q: Why did you check the solutions?

A: We checked the solutions to make sure they were valid. This involved plugging the solutions back into the original equation to see if they were true.

Q: What if the quadratic equation had not factored easily?

A: If the quadratic equation had not factored easily, we could have used other methods to solve it, such as the quadratic formula. The quadratic formula is a formula that can be used to solve quadratic equations of the form ax^2 + bx + c = 0.

Q: Can you explain the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations of the form ax^2 + bx + c = 0. The formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

This formula can be used to find the solutions to a quadratic equation.

Q: What if the equation had been more complex?

A: If the equation had been more complex, we would have needed to use more advanced techniques to solve it. This could have involved using algebraic manipulations, such as multiplying both sides of the equation by a common factor, or using other mathematical techniques, such as calculus.

Conclusion

In this article, we answered some frequently asked questions related to solving for x in the equation x+7βˆ’2=xβˆ’1\sqrt{x+7} - 2 = x - 1. We used algebraic manipulations to isolate the variable x and find its value. We also discussed some of the techniques that can be used to solve more complex equations.

Additional Resources

For more information on solving quadratic equations, see the following resources:

Final Answer

The final answer is x = -3 or x = 2.