Solve The Equation By The Square Root Property: $(x+5)^2=16$The Solution Set Is $\{\ \square\ \}$. (Simplify Your Answer. Type An Exact Answer, Using Radicals If Necessary.)

Introduction

The square root property is a powerful tool in algebra that allows us to solve equations of the form $(x-a)^2=b$. This property states that if $(x-a)^2=b$, then $x-a=\pm\sqrt{b}$, and therefore $x=a\pm\sqrt{b}$. In this article, we will explore how to solve equations by the square root property, using the equation $(x+5)^2=16$ as an example.

The Square Root Property

The square root property is based on the fact that the square of a number is always non-negative. In other words, if $x$ is a real number, then $x^2\geq 0$. This means that if we have an equation of the form $(x-a)^2=b$, we can always find a value of $x$ that satisfies the equation.

To see why this is the case, let's consider the equation $(x-a)^2=b$. We can rewrite this equation as $x^2-2ax+a^2=b$. This is a quadratic equation in $x$, and we can solve it using the quadratic formula.

However, there is a much simpler way to solve this equation. Since $x^2-2ax+a^2=b$, we know that $x^2-2ax+a^2-b=0$. This is a quadratic equation in $x$, and we can factor it as $(x-a)^2-b=0$.

Now, we can take the square root of both sides of this equation to get $x-a=\pm\sqrt{b}$. This gives us two possible values for $x$: $x=a+\sqrt{b}$ and $x=a-\sqrt{b}$.

Solving the Equation $(x+5)^2=16$

Now that we have a good understanding of the square root property, let's use it to solve the equation $(x+5)^2=16$. This equation is in the form $(x-a)^2=b$, where $a=-5$ and $b=16$.

Using the square root property, we can rewrite this equation as $x+5=\pm\sqrt{16}$. This gives us two possible values for $x$: $x+5=4$ and $x+5=-4$.

Solving for $x$ in both cases, we get $x=-1$ and $x=-9$.

Conclusion

In this article, we have seen how to solve equations by the square root property. We have used the equation $(x+5)^2=16$ as an example, and have shown that the solution set is $\{-1,-9\}$.

The square root property is a powerful tool in algebra that allows us to solve equations of the form $(x-a)^2=b$. It is based on the fact that the square of a number is always non-negative, and it allows us to find a value of $x$ that satisfies the equation.

Example Problems

Here are a few example problems that you can try to solve using the square root property:

• $(x-3)^2=9$
• $(x+2)^2=25$
• $(x-1)^2=36$

Solutions

Here are the solutions to the example problems:

• $(x-3)^2=9$ $\Rightarrow$ $x-3=\pm\sqrt{9}$ $\Rightarrow$ $x=3\pm3$ $\Rightarrow$ $x=6$ or $x=0$
• $(x+2)^2=25$ $\Rightarrow$ $x+2=\pm\sqrt{25}$ $\Rightarrow$ $x=-2\pm5$ $\Rightarrow$ $x=3$ or $x=-7$
• $(x-1)^2=36$ $\Rightarrow$ $x-1=\pm\sqrt{36}$ $\Rightarrow$ $x=1\pm6$ $\Rightarrow$ $x=7$ or $x=-5$

Tips and Tricks

Here are a few tips and tricks that you can use to help you solve equations by the square root property:

• Make sure that the equation is in the form $(x-a)^2=b$.
• Use the square root property to rewrite the equation as $x-a=\pm\sqrt{b}$.
• Solve for $x$ in both cases.
• Check your solutions to make sure that they are correct.

Conclusion

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Introduction

In our previous article, we explored how to solve equations by the square root property. We used the equation $(x+5)^2=16$ as an example and showed that the solution set is $\{-1,-9\}$. In this article, we will answer some frequently asked questions about solving equations by the square root property.

Q: What is the square root property?

A: The square root property is a powerful tool in algebra that allows us to solve equations of the form $(x-a)^2=b$. It states that if $(x-a)^2=b$, then $x-a=\pm\sqrt{b}$, and therefore $x=a\pm\sqrt{b}$.

Q: How do I know if an equation can be solved using the square root property?

A: To determine if an equation can be solved using the square root property, you need to check if it is in the form $(x-a)^2=b$. If it is, then you can use the square root property to solve the equation.

Q: What if the equation has a negative number under the square root?

A: If the equation has a negative number under the square root, then it is not possible to find a real solution using the square root property. In this case, you may need to use other methods to solve the equation.

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

A: Yes, you can use the square root property to solve equations with fractions under the square root. However, you need to be careful when simplifying the expression under the square root.

Q: How do I simplify the expression under the square root?

A: To simplify the expression under the square root, you need to find the square root of the number under the square root. If the number under the square root is a perfect square, then you can simplify the expression by taking the square root of the perfect square.

Q: What if I get a negative number when simplifying the expression under the square root?

A: If you get a negative number when simplifying the expression under the square root, then you need to be careful when solving the equation. In this case, you may need to use other methods to solve the equation.

Q: Can I use the square root property to solve equations with variables under the square root?

A: Yes, you can use the square root property to solve equations with variables under the square root. However, you need to be careful when simplifying the expression under the square root.

Q: How do I solve equations with variables under the square root?

A: To solve equations with variables under the square root, you need to isolate the variable under the square root. Then, you can use the square root property to solve the equation.

Q: What if I get a complex number when solving the equation?

A: If you get a complex number when solving the equation, then you need to be careful when simplifying the expression. In this case, you may need to use other methods to solve the equation.

Conclusion

In this article, we have answered some frequently asked questions about solving equations by the square root property. We have provided examples and explanations to help you understand how to use the square root property to solve equations. We hope that this article has been helpful in clarifying any confusion you may have had about solving equations by the square root property.

Example Problems

Here are a few example problems that you can try to solve using the square root property:

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

Solutions

Here are the solutions to the example problems:

• $(x-2)^2=9$ $\Rightarrow$ $x-2=\pm\sqrt{9}$ $\Rightarrow$ $x=2\pm3$ $\Rightarrow$ $x=5$ or $x=-1$
• $(x+3)^2=16$ $\Rightarrow$ $x+3=\pm\sqrt{16}$ $\Rightarrow$ $x=-3\pm4$ $\Rightarrow$ $x=1$ or $x=-7$
• $(x-4)^2=25$ $\Rightarrow$ $x-4=\pm\sqrt{25}$ $\Rightarrow$ $x=4\pm5$ $\Rightarrow$ $x=9$ or $x=-1$

Tips and Tricks

Here are a few tips and tricks that you can use to help you solve equations by the square root property:

• Make sure that the equation is in the form $(x-a)^2=b$.
• Use the square root property to rewrite the equation as $x-a=\pm\sqrt{b}$.
• Solve for $x$ in both cases.
• Check your solutions to make sure that they are correct.
• Be careful when simplifying the expression under the square root.
• Use other methods to solve the equation if you get a complex number or a negative number under the square root.