Okay, so this is a high school level assignment:
$$
\sqrt{x+14}-\sqrt{x+5}=\sqrt{x-2}-\sqrt{x-7}
$$
Here's a similar one:
$$
\sqrt{x}+\sqrt{x-5}=\sqrt{x+7}+\sqrt{x-8}
$$
When solving these traditionaly, I get a polynomial with exponent to the 4th which I cannot solve (I can guess the solutions via free term and divide the polynomial, accordingly, but I don't think that is the intended method here).
Is there a trick to any of these two tasks that would prevent exponents from getting out of control? How would a highschooler solve them?
Best Answer
write like this
$$\sqrt{x+14}+\sqrt{x-7}=\sqrt{x-2}+\sqrt{x+5}$$
square both sides:
$$2x+7+2\sqrt{(x+14)(x-7)}=2x+3+2\sqrt{(x-2)(x+5)}\\ 2+\sqrt{(x+14)(x-7)}=\sqrt{(x-2)(x+5)}$$
square again
$$4\sqrt{(x+14)(x-7)}+x^2+7x-94=x^2+3x-10\\ \sqrt{(x+14)(x-7)}=21-x \to (x+14)(x-7)=(21-x)^2$$
Can you finish?
Don't forget to test the result in the original equation.