### Instruction: Find the positive real number $x$ for which
\[5 \sqrt{1 + x} + 5 \sqrt{1 - x} = 7 \sqrt{2}.\].### Response: This problem involves square roots and an equation, so I wonder if I can get rid of the square roots by squaring both sides.
Before I do that, I notice that the equation only makes sense if $x$ is between -1 and 1, because otherwise the square roots would not be real.
So I already know that $x$ is in the interval $(-1, 1)$.
Squaring both sides gives me
\[25(1 + x) + 25(1 - x) + 50 \sqrt{(1 + x)(1 - x)} = 98.\]
Simplifying a bit, I get
\[50 + 50 \sqrt{1 - x^2} = 98.\]
Subtracting 50 from both sides and dividing by 50, I get
\[\sqrt{1 - x^2} = 0.96.\]
Now I only have one square root left, so I can square both sides again and get
\[1 - x^2 = 0.9216.\]
Subtracting 0.9216 from both sides and multiplying by -1, I get
\[x^2 = 0.0784.\]
Taking the square root of both sides, I get
\[x = \pm \sqrt{0.0784}.\]
But I remember that $x$ has to be positive, so I only keep the positive root and get
\[x = \sqrt{0.0784}.\]
This is an exact answer, but I can also approximate it with a calculator if I want to.
I get
\[x \approx 0.28.\]