What Are The Solutions To The Equation X 2 + 2 X − 10 = X + 20 \sqrt{x^2+2x-10}=\sqrt{x+20} X 2 + 2 X − 10 ​ = X + 20 ​ ?A. X = − 6 , X = − 5 X=-6, X=-5 X = − 6 , X = − 5 B. X = − 6 , X = 5 X=-6, X=5 X = − 6 , X = 5 C. X = − 5 , X = 6 X=-5, X=6 X = − 5 , X = 6 D. X = 5 , X = 6 X=5, X=6 X = 5 , X = 6

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Introduction

Solving equations involving square roots can be a challenging task, especially when the equations are complex and involve multiple variables. In this article, we will explore the solutions to the equation $\sqrt{x^2+2x-10}=\sqrt{x+20}$, and we will examine the different methods that can be used to solve this type of equation.

Understanding the Equation

The given equation is $\sqrt{x^2+2x-10}=\sqrt{x+20}$. To solve this equation, we need to isolate the variable $x$ and eliminate the square roots. We can start by squaring both sides of the equation, which will allow us to eliminate the square roots and simplify the equation.

Squaring Both Sides

When we square both sides of the equation, we get:

$$x^2+2x-10=x+20$$

This equation is now a quadratic equation, and we can use the standard methods for solving quadratic equations to find the solutions.

Expanding and Simplifying

To simplify the equation, we can expand the left-hand side and combine like terms:

$$x^2+2x-10=x+20$$

$$x^2+x-30=0$$

Factoring the Quadratic Equation

The quadratic equation $x^2+x-30=0$ can be factored as:

$$(x+6)(x-5)=0$$

Solving for $x$

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

$$x+6=0 \Rightarrow x=-6$$

$$x-5=0 \Rightarrow x=5$$

Checking the Solutions

To check the solutions, we can substitute each value of $x$ back into the original equation and verify that it is true. If the equation is true, then the value of $x$ is a valid solution.

Substituting $x=-6$

When we substitute $x=-6$ into the original equation, we get:

$$\sqrt{(-6)^2+2(-6)-10}=\sqrt{-6+20}$$

$$\sqrt{36-12-10}=\sqrt{4}$$

$$\sqrt{14}=\sqrt{4}$$

This equation is not true, so $x=-6$ is not a valid solution.

Substituting $x=5$

When we substitute $x=5$ into the original equation, we get:

$$\sqrt{5^2+2(5)-10}=\sqrt{5+20}$$

$$\sqrt{25+10-10}=\sqrt{25}$$

$$\sqrt{25}=\sqrt{25}$$

This equation is true, so $x=5$ is a valid solution.

Conclusion

In this article, we have explored the solutions to the equation $\sqrt{x^2+2x-10}=\sqrt{x+20}$. We have used the method of squaring both sides to eliminate the square roots and simplify the equation. We have also factored the quadratic equation and solved for $x$. Finally, we have checked the solutions by substituting each value of $x$ back into the original equation. The only valid solution to the equation is $x=5$.

Final Answer

The final answer is $\boxed{x=5}$.

Discussion

The equation $\sqrt{x^2+2x-10}=\sqrt{x+20}$ is a classic example of an equation involving square roots. The method of squaring both sides is a powerful tool for solving this type of equation, and it can be used to solve many other equations involving square roots. In this article, we have shown that the only valid solution to the equation is $x=5$. We hope that this article has been helpful in understanding the solutions to this type of equation.

Related Topics

• Solving quadratic equations
• Eliminating square roots
• Factoring quadratic equations
• Checking solutions

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman

Keywords

• Square roots
• Quadratic equations
• Factoring
• Checking solutions
• Algebra
• Calculus
• Mathematics for computer science
  

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Introduction

In our previous article, we explored the solutions to the equation $\sqrt{x^2+2x-10}=\sqrt{x+20}$. We used the method of squaring both sides to eliminate the square roots and simplify the equation. We also factored the quadratic equation and solved for $x$. Finally, we checked the solutions by substituting each value of $x$ back into the original equation. In this article, we will answer some of the most frequently asked questions about the solutions to this equation.

Q: What is the final answer to the equation $\sqrt{x^2+2x-10}=\sqrt{x+20}$?

A: The final answer to the equation $\sqrt{x^2+2x-10}=\sqrt{x+20}$ is $\boxed{x=5}$.

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

A: We squared both sides of the equation to eliminate the square roots and simplify the equation. This allowed us to use the standard methods for solving quadratic equations to find the solutions.

Q: How did we factor the quadratic equation?

A: We factored the quadratic equation $x^2+x-30=0$ as $(x+6)(x-5)=0$.

Q: Why did we check the solutions?

A: We checked the solutions to verify that they were valid. If the equation is true, then the value of $x$ is a valid solution.

Q: What is the significance of the solutions to this equation?

A: The solutions to this equation are significant because they demonstrate the importance of checking solutions. If we had not checked the solutions, we would have concluded that $x=-6$ was a valid solution, which is not true.

Q: Can we use the method of squaring both sides to solve other equations involving square roots?

A: Yes, we can use the method of squaring both sides to solve other equations involving square roots. This method is a powerful tool for solving this type of equation.

Q: What are some other methods for solving equations involving square roots?

A: Some other methods for solving equations involving square roots include using the conjugate, using the Pythagorean theorem, and using algebraic manipulations.

Q: How can we use the solutions to this equation in real-world applications?

A: The solutions to this equation can be used in real-world applications such as physics, engineering, and computer science. For example, we can use the solutions to model the motion of an object under the influence of gravity.

Q: What are some common mistakes to avoid when solving equations involving square roots?

A: Some common mistakes to avoid when solving equations involving square roots include failing to check solutions, failing to simplify the equation, and using the wrong method to solve the equation.

Q: How can we use technology to solve equations involving square roots?

A: We can use technology such as calculators and computer software to solve equations involving square roots. These tools can help us to simplify the equation and find the solutions.

Q: What are some other resources for learning about solving equations involving square roots?

A: Some other resources for learning about solving equations involving square roots include textbooks, online tutorials, and video lectures.

Conclusion

In this article, we have answered some of the most frequently asked questions about the solutions to the equation $\sqrt{x^2+2x-10}=\sqrt{x+20}$. We have demonstrated the importance of checking solutions and using the method of squaring both sides to eliminate the square roots and simplify the equation. We have also discussed some other methods for solving equations involving square roots and provided some resources for learning about this topic.

Final Answer

The final answer is $\boxed{x=5}$.

Discussion

The equation $\sqrt{x^2+2x-10}=\sqrt{x+20}$ is a classic example of an equation involving square roots. The method of squaring both sides is a powerful tool for solving this type of equation, and it can be used to solve many other equations involving square roots. In this article, we have shown that the only valid solution to the equation is $x=5$. We hope that this article has been helpful in understanding the solutions to this type of equation.

Related Topics

• Solving quadratic equations
• Eliminating square roots
• Factoring quadratic equations
• Checking solutions
• Algebra
• Calculus
• Mathematics for computer science

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman

Keywords

• Square roots
• Quadratic equations
• Factoring
• Checking solutions
• Algebra
• Calculus
• Mathematics for computer science