Which Equation Results From Taking The Square Root Of Both Sides Of $(x+9)^2=25$?A. $x+3= \pm 5$B. $x+3= \pm 25$C. $x+9= \pm 5$D. $x+9= \pm 25$

Introduction

In mathematics, equations are a fundamental concept that helps us solve problems and understand various mathematical relationships. One of the most common techniques used to solve equations is by taking the square root of both sides. In this article, we will explore how to take the square root of both sides of an equation and provide a step-by-step guide on how to solve the resulting equation.

Understanding the Given Equation

The given equation is $(x+9)^2=25$. This equation represents a quadratic equation in the form of $(x+a)^2=b$, where $a$ and $b$ are constants. To solve this equation, we need to take the square root of both sides.

Taking the Square Root of Both Sides

When we take the square root of both sides of an equation, we are essentially finding the square root of the expression on the left-hand side and the right-hand side separately. In this case, we have:

$$(x+9)^2=25$$

Taking the square root of both sides gives us:

$$\sqrt{(x+9)^2}=\sqrt{25}$$

Using the property of square roots, we can simplify the left-hand side as:

$$|x+9|=\sqrt{25}$$

Simplifying the Right-Hand Side

The right-hand side of the equation is $\sqrt{25}$. Since the square root of 25 is 5, we can simplify the right-hand side as:

$$|x+9|=5$$

Removing the Absolute Value

The absolute value of an expression is the distance of the expression from zero on the number line. In this case, we have:

$$|x+9|=5$$

To remove the absolute value, we need to consider two cases:

Case 1: $x+9 \geq 0$

In this case, the absolute value can be written as:

$$x+9=5$$

Case 2: $x+9 < 0$

In this case, the absolute value can be written as:

$$x+9=-5$$

Solving for x

Now that we have removed the absolute value, we can solve for $x$ in each case.

Case 1: $x+9=5$

Subtracting 9 from both sides gives us:

$$x=-4$$

Case 2: $x+9=-5$

Subtracting 9 from both sides gives us:

$$x=-14$$

Conclusion

In this article, we have explored how to take the square root of both sides of an equation and solve the resulting equation. We have used the given equation $(x+9)^2=25$ as an example and have shown how to remove the absolute value and solve for $x$. The resulting equations are $x=-4$ and $x=-14$.

Answer

The correct answer is:

A. $x+3= \pm 5$

This is because we can rewrite the equation $x+9= \pm 5$ as $x+3= \pm 5-3+3$, which simplifies to $x+3= \pm 5$.

Discussion

The equation $(x+9)^2=25$ can be solved by taking the square root of both sides and removing the absolute value. The resulting equations are $x=-4$ and $x=-14$. This problem requires a good understanding of quadratic equations and the properties of square roots.

Additional Resources

For more information on solving equations and quadratic equations, please refer to the following resources:

• Khan Academy: Solving Quadratic Equations
• Mathway: Solving Quadratic Equations
• Wolfram Alpha: Solving Quadratic Equations

Final Thoughts

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Introduction

In our previous article, we explored how to take the square root of both sides of an equation and solve the resulting equation. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving equations.

Q: What is the first step in solving an equation?

A: The first step in solving an equation is to read and understand the equation. This involves identifying the variables, constants, and any mathematical operations involved.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, whereas a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the following steps:

1. Add or subtract the same value to both sides of the equation to isolate the variable.
2. Multiply or divide both sides of the equation by the same value to isolate the variable.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

1. Factor the quadratic expression, if possible.
2. Use the quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic expression.
3. Use the method of completing the square to solve the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula used to solve quadratic equations. It is given by:

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Q: What is the method of completing the square?

A: The method of completing the square is a technique used to solve quadratic equations. It involves rewriting the quadratic expression in the form $(x+a)^2=b$, where $a$ and $b$ are constants.

Q: How do I take the square root of both sides of an equation?

A: To take the square root of both sides of an equation, you can use the following steps:

1. Check if the equation is a perfect square trinomial.
2. If it is, rewrite the equation in the form $(x+a)^2=b$.
3. Take the square root of both sides of the equation.

Q: What is the difference between a positive and negative square root?

A: A positive square root is a square root that is equal to the original value, whereas a negative square root is a square root that is equal to the negative of the original value.

Q: How do I remove the absolute value from an equation?

A: To remove the absolute value from an equation, you can use the following steps:

1. Check if the expression inside the absolute value is positive or negative.
2. If it is positive, rewrite the equation as $x+a=b$.
3. If it is negative, rewrite the equation as $x+a=-b$.

Q: What is the final step in solving an equation?

A: The final step in solving an equation is to check your solution by plugging it back into the original equation.

Conclusion

Solving equations is an essential skill in mathematics, and understanding the concepts and techniques involved can help you solve equations with confidence. By following the steps outlined in this Q&A guide, you can better understand how to solve equations and improve your problem-solving skills.

Additional Resources

For more information on solving equations and quadratic equations, please refer to the following resources:

• Khan Academy: Solving Quadratic Equations
• Mathway: Solving Quadratic Equations
• Wolfram Alpha: Solving Quadratic Equations

Final Thoughts

Solving equations is a fundamental skill in mathematics, and understanding the concepts and techniques involved can help you solve equations with confidence. By following the steps outlined in this Q&A guide, you can improve your problem-solving skills and become a more confident math student.