\begin{tabular}{|l|l|}\hline \multicolumn{1}{|c|}{STEPS:} & $(x+20)^{\frac{1}{2}}=x$ \\\hline 1. Isolate The Power & \\\hline \begin{tabular}{l} 2. Eliminate The Power By Raising Each Side Of \\ The Equation To The Reciprocal Of The

Solving the Equation: $(x+20)^{\frac{1}{2}}=x$

In mathematics, solving equations with exponents and radicals is a crucial skill that requires careful attention to detail and a solid understanding of algebraic concepts. The equation $(x+20)^{\frac{1}{2}}=x$ is a classic example of an equation that involves a square root and a linear term. In this article, we will guide you through the step-by-step process of solving this equation, and provide you with a clear understanding of the underlying mathematical concepts.

Step 1: Isolate the Power

The first step in solving the equation is to isolate the power. In this case, the power is the square root of $(x+20)$. To isolate the power, we need to get rid of the $x$ term on the right-hand side of the equation. We can do this by subtracting $x$ from both sides of the equation.

(x+20)^{\frac{1}{2}} - x = 0

This gives us a new equation that has the power isolated on one side.

Step 2: Eliminate the Power

Now that we have isolated the power, we need to eliminate it by raising each side of the equation to the reciprocal of the exponent. In this case, the reciprocal of the exponent is $2$, since the exponent is $\frac{1}{2}$. To eliminate the power, we need to square both sides of the equation.

(x+20)^{\frac{1}{2}} = x
(x+20)^{\frac{1}{2}} - x = 0
(x+20)^{\frac{1}{2}} - x)^2 = 0^2
(x+20) - 2x(x+20)^{\frac{1}{2}} + x^2 = 0

This gives us a new equation that has the power eliminated.

Simplifying the Equation

Now that we have eliminated the power, we can simplify the equation by combining like terms. We can start by expanding the squared term on the left-hand side of the equation.

(x+20) - 2x(x+20)^{\frac{1}{2}} + x^2 = 0
x^2 + 20x - 2x(x+20)^{\frac{1}{2}} + 20 = 0

Next, we can combine like terms by adding or subtracting the coefficients of the same variables.

x^2 + 20x - 2x(x+20)^{\frac{1}{2}} + 20 = 0
x^2 + 20x - 2x(x+20)^{\frac{1}{2}} = -20

Rearranging the Equation

Now that we have simplified the equation, we can rearrange it to make it easier to solve. We can start by moving all the terms to one side of the equation.

x^2 + 20x - 2x(x+20)^{\frac{1}{2}} = -20
x^2 + 20x - 2x(x+20)^{\frac{1}{2}} + 20 = 0

This gives us a new equation that has all the terms on one side.

Solving the Equation

Now that we have rearranged the equation, we can solve it by factoring or using other algebraic techniques. In this case, we can use the quadratic formula to solve the equation.

x^2 + 20x - 2x(x+20)^{\frac{1}{2}} + 20 = 0
x^2 + 20x - 2x(x+20)^{\frac{1}{2}} = -20
x^2 + 20x - 2x(x+20)^{\frac{1}{2}} + 20 = 0

The quadratic formula is given by:

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

In this case, $a = 1$, $b = 20$, and $c = -2x(x+20)^{\frac{1}{2}} + 20$. Plugging these values into the quadratic formula, we get:

x = \frac{-20 \pm \sqrt{20^2 - 4(1)(-2x(x+20)^{\frac{1}{2}} + 20)}}{2(1)}

Simplifying the expression under the square root, we get:

x = \frac{-20 \pm \sqrt{400 + 8x(x+20)^{\frac{1}{2}} - 80}}{2}

This gives us two possible solutions for $x$.

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In the previous article, we solved the equation $(x+20)^{\frac{1}{2}}=x$ using a step-by-step approach. However, we know that there are many questions that readers may have about this equation and its solution. In this article, we will answer some of the most frequently asked questions about solving the equation $(x+20)^{\frac{1}{2}}=x$.

Q: What is the first step in solving the equation $(x+20)^{\frac{1}{2}}=x$?

A: The first step in solving the equation $(x+20)^{\frac{1}{2}}=x$ is to isolate the power. In this case, the power is the square root of $(x+20)$. To isolate the power, we need to get rid of the $x$ term on the right-hand side of the equation. We can do this by subtracting $x$ from both sides of the equation.

Q: How do I eliminate the power in the equation $(x+20)^{\frac{1}{2}}=x$?

A: To eliminate the power, we need to raise each side of the equation to the reciprocal of the exponent. In this case, the reciprocal of the exponent is $2$, since the exponent is $\frac{1}{2}$. To eliminate the power, we need to square both sides of the equation.

Q: What is the quadratic formula, and how do I use it to solve the equation $(x+20)^{\frac{1}{2}}=x$?

A: The quadratic formula is given by:

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

In this case, $a = 1$, $b = 20$, and $c = -2x(x+20)^{\frac{1}{2}} + 20$. Plugging these values into the quadratic formula, we get:

x = \frac{-20 \pm \sqrt{20^2 - 4(1)(-2x(x+20)^{\frac{1}{2}} + 20)}}{2(1)}

Simplifying the expression under the square root, we get:

x = \frac{-20 \pm \sqrt{400 + 8x(x+20)^{\frac{1}{2}} - 80}}{2}

This gives us two possible solutions for $x$.

Q: What are the two possible solutions for $x$ in the equation $(x+20)^{\frac{1}{2}}=x$?

A: The two possible solutions for $x$ in the equation $(x+20)^{\frac{1}{2}}=x$ are given by:

x = \frac{-20 \pm \sqrt{400 + 8x(x+20)^{\frac{1}{2}} - 80}}{2}

These solutions can be simplified to:

x = \frac{-20 \pm \sqrt{320 + 8x(x+20)^{\frac{1}{2}}}}{2}

Q: How do I check my solutions to the equation $(x+20)^{\frac{1}{2}}=x$?

A: To check your solutions, you need to plug them back into the original equation and verify that they satisfy the equation. In this case, you need to plug the two possible solutions for $x$ back into the equation $(x+20)^{\frac{1}{2}}=x$ and verify that they satisfy the equation.

Q: What are some common mistakes to avoid when solving the equation $(x+20)^{\frac{1}{2}}=x$?

A: Some common mistakes to avoid when solving the equation $(x+20)^{\frac{1}{2}}=x$ include:

• Not isolating the power correctly
• Not eliminating the power correctly
• Not using the quadratic formula correctly
• Not checking the solutions correctly

By avoiding these common mistakes, you can ensure that you solve the equation correctly and obtain the correct solutions.

In this article, we have answered some of the most frequently asked questions about solving the equation $(x+20)^{\frac{1}{2}}=x$. We have provided step-by-step instructions on how to solve the equation, and have also provided some common mistakes to avoid. By following these instructions and avoiding these common mistakes, you can ensure that you solve the equation correctly and obtain the correct solutions.