Find the exact solution for each equation. Find the approximate solution to the nearest tenth.
5 sq rt 2/ sq rt 2 -1 = x / sq rt 2
Please explain in detail how to work this. I am really confused. Thanks
5√2/(√2-1) = x/√2
5√2√2/(√2-1) = x
x = 10/(√2-1)
rationalize the denominator to get
x = 10(√2+1)
To solve this equation, we need to isolate the variable x on one side of the equation. Let's break down the process step by step:
Step 1: Simplify the equation if possible.
The equation is:
(5√2 / √2 - 1) = x / √2
First, simplify the expression inside the parentheses by rationalizing the denominator (√2 - 1).
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of √2 - 1, which is √2 + 1:
(5√2 / (√2 - 1)) * (√2 + 1) / (√2 + 1) = x / √2
Simplifying further, we have:
(5√2 * (√2 + 1)) / ((√2 - 1) * (√2 + 1)) = x / √2
Expanding the numerator and denominator:
(5√2 * √2 + 5√2) / (√2 * √2 - 1 * √2 + √2 - 1)
(10 + 5√2) / (2 - 1) = x / √2
Simplifying:
(10 + 5√2) = x / √2
Step 2: Isolate the variable x.
To isolate x, we need to multiply both sides of the equation by √2:
(10 + 5√2) * √2 = x
Simplifying further:
10√2 + 5 * 2 = x
20 + 10√2 = x
The exact solution to the equation is x = 20 + 10√2.
Step 3: Find the approximate solution to the nearest tenth.
To approximate the solution, we can substitute the value of √2 ≈ 1.414 into the equation:
x ≈ 20 + 10 * 1.414
x ≈ 20 + 14.14
x ≈ 34.14
So, the approximate solution to the nearest tenth is x ≈ 34.1.