Solve $90=(x-4)^2$, Where $x$ Is A Real Number. Round Your Answer To The Nearest Hundredth.If There Is More Than One Solution, Separate Them With Commas. If There Is No Solution, Click On No Solution.
Introduction
In this article, we will be solving the quadratic equation $90=(x-4)^2$, where $x$ is a real number. This equation is a quadratic equation in the form of $a(x-h)^2=k$, where $a$, $h$, and $k$ are constants. Our goal is to find the value of $x$ that satisfies this equation.
Understanding the Equation
The given equation is $90=(x-4)^2$. This equation can be rewritten as $(x-4)^2=90$. To solve for $x$, we need to isolate $x$ on one side of the equation. We can do this by taking the square root of both sides of the equation.
Taking the Square Root of Both Sides
Taking the square root of both sides of the equation gives us $x-4=\pm\sqrt90}$. This equation has two possible solutions$ and $x-4=-\sqrt{90}$.
Solving for $x$
To solve for $x$, we need to isolate $x$ on one side of the equation. We can do this by adding $4$ to both sides of the equation.
Solution 1: $x-4=\sqrt{90}$
Adding $4$ to both sides of the equation gives us $x=\sqrt{90}+4$. To simplify this expression, we can evaluate the square root of $90$.
Substituting this value back into the equation gives us $x=3\sqrt{10}+4$.
Solution 2: $x-4=-\sqrt{90}$
Adding $4$ to both sides of the equation gives us $x=-\sqrt{90}+4$. To simplify this expression, we can evaluate the square root of $90$.
Substituting this value back into the equation gives us $x=-3\sqrt{10}+4$.
Evaluating the Solutions
To evaluate the solutions, we need to round them to the nearest hundredth.
Solution 1: $x=3\sqrt{10}+4$
Evaluating the square root of $10$ gives us $\sqrt{10}\approx3.16$. Substituting this value back into the equation gives us $x=3(3.16)+4\approx9.48+4\approx13.48$.
Solution 2: $x=-3\sqrt{10}+4$
Evaluating the square root of $10$ gives us $\sqrt{10}\approx3.16$. Substituting this value back into the equation gives us $x=-3(3.16)+4\approx-9.48+4\approx-5.48$.
Conclusion
In conclusion, the solutions to the equation $90=(x-4)^2$ are $x\approx13.48$ and $x\approx-5.48$. These solutions are rounded to the nearest hundredth.
Final Answer
The final answer is:
Introduction
In this article, we will be answering some of the most frequently asked questions about solving the quadratic equation $90=(x-4)^2$. This equation is a quadratic equation in the form of $a(x-h)^2=k$, where $a$, $h$, and $k$ are constants. Our goal is to provide a clear and concise explanation of the solution process and address any common misconceptions.
Q: What is the first step in solving the equation $90=(x-4)^2$?
A: The first step in solving the equation $90=(x-4)^2$ is to take the square root of both sides of the equation. This gives us $x-4=\pm\sqrt{90}$.
Q: Why do we need to take the square root of both sides of the equation?
A: We need to take the square root of both sides of the equation because the square root is the inverse operation of squaring. By taking the square root of both sides, we can isolate $x$ on one side of the equation.
Q: What is the difference between $x-4=\sqrt{90}$ and $x-4=-\sqrt{90}$?
A: The difference between $x-4=\sqrt{90}$ and $x-4=-\sqrt{90}$ is that the first equation has a positive square root, while the second equation has a negative square root. This means that the first equation has a solution that is greater than $4$, while the second equation has a solution that is less than $4$.
Q: How do we evaluate the solutions to the equation $90=(x-4)^2$?
A: To evaluate the solutions to the equation $90=(x-4)^2$, we need to round them to the nearest hundredth. This means that we need to use a calculator to evaluate the square root of $90$ and then round the result to the nearest hundredth.
Q: What is the final answer to the equation $90=(x-4)^2$?
A: The final answer to the equation $90=(x-4)^2$ is $x\approx13.48$ and $x\approx-5.48$. These solutions are rounded to the nearest hundredth.
Q: Can we simplify the solutions to the equation $90=(x-4)^2$?
A: Yes, we can simplify the solutions to the equation $90=(x-4)^2$ by evaluating the square root of $90$ and then simplifying the resulting expression. For example, we can simplify $x=3\sqrt{10}+4$ by evaluating the square root of $10$ and then simplifying the resulting expression.
Q: What is the significance of the equation $90=(x-4)^2$?
A: The equation $90=(x-4)^2$ is a quadratic equation in the form of $a(x-h)^2=k$, where $a$, $h$, and $k$ are constants. This equation has two solutions, which are $x\approx13.48$ and $x\approx-5.48$. The significance of this equation is that it provides a clear and concise example of how to solve a quadratic equation.
Q: Can we use the equation $90=(x-4)^2$ to model real-world problems?
A: Yes, we can use the equation $90=(x-4)^2$ to model real-world problems. For example, we can use this equation to model the motion of an object that is moving in a straight line. The equation $90=(x-4)^2$ can be used to model the position of the object at any given time.
Q: What are some common mistakes to avoid when solving the equation $90=(x-4)^2$?
A: Some common mistakes to avoid when solving the equation $90=(x-4)^2$ include:
- Not taking the square root of both sides of the equation
- Not isolating $x$ on one side of the equation
- Not rounding the solutions to the nearest hundredth
- Not simplifying the solutions to the equation
Conclusion
In conclusion, the equation $90=(x-4)^2$ is a quadratic equation that has two solutions, which are $x\approx13.48$ and $x\approx-5.48$. These solutions are rounded to the nearest hundredth. We hope that this article has provided a clear and concise explanation of the solution process and addressed any common misconceptions.