3.29x=5.5√(x)+10,000
How in the world do I solve for x without using a calculator? I swear I've been on this for hours. I know the answer is 3133 because I estimated the answer and then began plugging in numbers until I got to the actual answer.
If the question says to solve without a calculator, then admittedly this is a very unreasonable question.
Not even in my olden days, (you know when we had to rely on slide rules and log tables) would they have expected a solution.
I ran it through Wolfram and it shows that you were close,
http://www.wolframalpha.com/input/?i=solve+3.29x%3D5.5%E2%88%9A%28x%29%2B10%2C000
x = appr 3133.1
If you really had to do this .....
3.29x - 10000 = 5.5√x
now square both sides and you would get a quadratic with messy numbers
use the quadratic formula (? with those messy numbers and without a calculator ??) to get two answers.
both must be checked, since we squared our equation.
You would find that the 3133.1 works and the other one does not
It didn't say solve without, I just really wanted to figure out a way to show work and to keep my self esteem intact. I did the squaring on both sides, moved everything to one side and made the other side be 0, and used quadratic equation. Thanks ^.^
Really?
Then what does this mean ?
"How in the world do I solve for x without using a calculator?"
It means I wanted to. I really like math and don't enjoy using calculator. Don't get inquisitorial about it, doesn't really matter.
To solve the equation 3.29x = 5.5√(x) + 10,000 without using a calculator, you can follow these steps:
Step 1: Isolate the square root term.
- Subtract 5.5√(x) from both sides of the equation to get: 3.29x - 5.5√(x) = 10,000
Step 2: Eliminate the square root term.
- Square both sides of the equation to eliminate the square root: (3.29x - 5.5√(x))^2 = (10,000)^2
Step 3: Expand and simplify the left side of the equation.
- Remember that (a - b)^2 is equivalent to a^2 - 2ab + b^2.
- Expanding and simplifying the left side gives you: (3.29x)^2 - 2(3.29x)(5.5√(x)) + (5.5√(x))^2 = 100,000,000
Step 4: Simplify terms.
- The terms (3.29x)^2 and (5.5√(x))^2 can be simplified:
(3.29x)^2 = 10.8241x^2
(5.5√(x))^2 = 30.25x
The equation becomes: 10.8241x^2 - 2(3.29x)(5.5√(x)) + 30.25x = 100,000,000
Step 5: Continue simplifying.
- Distribute the 2(3.29x)(5.5√(x)) term: 10.8241x^2 - 2(3.29)(5.5)(√(x))^2 + 30.25x = 100,000,000
- Simplify the multiplied constants: 10.8241x^2 - 36.047x√(x) + 30.25x = 100,000,000
Step 6: Simplify further if possible.
- Simplify the x terms by factoring out x: x(10.8241x - 36.047√(x) + 30.25) = 100,000,000
Step 7: Solve for x.
- Divide both sides of the equation by (10.8241x - 36.047√(x) + 30.25): x = 100,000,000 / (10.8241x - 36.047√(x) + 30.25)
Unfortunately, at this point, it is difficult to isolate x algebraically without using a calculator. Hence, you may resort to approximation methods or numerical techniques to find an estimate for x.
Based on your estimation and plugging in numbers until reaching the actual answer, you found that the solution is x = 3133. Please note that this is an estimated solution and not the exact value. To obtain a more precise solution, you may need to use a calculator or numerical methods.