Question

I've written my question in the first reply to this. I'm new here.. looking for help with an additional mathematics course for GCSE's.

Answer 1

Q. Use suitable substitution and solve for 'x'.
\[2e ^{x} = 7 \sqrt{e ^{x}} - 3\]

Answer 2

For future reference, asking this question in the mathematics section would get a better response because solving \[2e ^{x} = 7 \sqrt{e ^{x}} - 3\] doesn't necessarily involves linear algebra. http://en.wikipedia.org/wiki/Linear_algebra

This section is primarily for linear algebra questions, that said, here is how to get started solving this problem.
\[ Let \;\; e^x = v\\2v = 7\sqrt{v} - 3 \]
Now, this equation can be factored like so \[0 = 2v-7\sqrt{v} +3\\ 0 = (2\sqrt{v}-1)(\sqrt{v}-3)\] Then, you can get the roots for v, and substitute back in \[e^x=v\]... let us know if any part of that is confusing...

If working with square roots is uncomfortable, you could isolate the square root, then square both sides. Then you will have a fairly routine looking quadratic to find the roots for.
\[2v = 7\sqrt{v} - 3\\ 2v+3 = 7\sqrt{v}\\4v^2+12v+9=49v\\4v^2-37v+9 =0 \]

And of course, don't forget that the quadratic formula can be used to find the roots, if factoring is not obvious.