To solve the given quadratic equation \((x - 3)^{2/5} = 20\), let's break down your questions step by step.
1. What should be the first step when solving the quadratic equation above?
The first step in solving the equation \((x - 3)^{2/5} = 20\) is to eliminate the fractional exponent. You can do this by raising both sides of the equation to the power of \(\frac{5}{2}\) to isolate \(x - 3\):
\[ ((x - 3)^{2/5})^{\frac{5}{2}} = 20^{\frac{5}{2}} \]
This simplifies to:
\[ x - 3 = 20^{\frac{5}{2}} \]
2. Which of the following is one of the solutions to the quadratic equation above?
Let's first calculate \(20^{\frac{5}{2}}\):
\[ 20^{\frac{5}{2}} = (20^5)^{1/2} = \sqrt{20^5} \]
Calculating \(20^5\) and finding its square root can be cumbersome, but the key idea is to focus on the expression \((x - 3) = 20^{\frac{5}{2}}\).
Once you determine \(x - 3\), you would find \(x\):
\[ x = 3 + 20^{\frac{5}{2}} \text{ (this is one solution)} \]
3. Which of the following is the other solution to the quadratic equation above?
The equation \((x - 3)^{2/5} = 20\) can have two potential cases since the 5th root can be both positive and negative when squared. So, we also consider:
\[ (x - 3) = -20^{\frac{5}{2}} \]
Thus, the second solution is found by solving:
\[ x - 3 = -20^{\frac{5}{2}} \]
So:
\[ x = 3 - 20^{\frac{5}{2}} \text{ (this is the other solution)} \]
Summary of Solutions
- The first step is to raise both sides of the equation to the power of \( \frac{5}{2} \).
- One solution is \( x = 3 + 20^{\frac{5}{2}} \).
- The other solution is \( x = 3 - 20^{\frac{5}{2}} \).
Please replace "20^5/2" with actual calculated numeric values if numeric answers are needed.
1 answer