Idiots Guide To Algebra

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I'm a complete idiot when it comes to algebra...Please help!?

I need help finding Z. No idea where to start.
√(-15z+361)-√(z+209)=-4
*Where Z is a real number*

Could anyone guide me through the whole process of finding Z so that next time I come across a problem like this, I can do it myself...
Thanks!

This is a very interesting and laborious problem, so please be patient with the solution

The answer is z = 16. Let us analyze as below.

Note : In problems involving square roots , it is best to approach them by using substitution methods to make the math simpler. This approach does involve a few steps so please be patient .

Step 1. Let √361 – 15z = a, and √z+ 209 = b
The problem can written as a - b = -4 – This is given to us - Equation 1
Now, let there be a number m such that a + b = m - Equation 2
Step 2. Now the simplification becomes easier. First we solve for m, then we will substitute the result for m in terms of z and finally solve for z.
Step 3:
Adding equations 1 and 2 above we get : 2a = m-4 – Equation 3
Subtracting equations 1 from Equation 2 we get : 2b = m +4 – Equation 4
Squaring Equation 3 on Both sides we get :
4a^2 = 4(361-15z) = m^2 -8m+16 – Equation 5
Squaring Equation 4 on Both sides we get :
4b^2 = 4 (z+209) = m^2 +8m+16 – Equation 6
Subtracting Equation 5 from Equation 6 and simplifying, we get: m = 4 z- 38
Step 4:
Now we should take Equation 3 to solve for z. (Because, arithmetically, Equation 4, returns imaginary roots , so it is ruled out)
2 a = 2(√361- 15z) = m-4 = 4z-38-4 = 4z - 42
Squaring on both sides , simplifying and and rearranging we get: 4z^2- 69z+80 = 0. Now we use the formula for quadratic equation to solve for z . This yields the roots of 16 and 1.25 for z.
Z =16 is the only root that satisfies the equations 1 and 2 (so we discard the other root with the value of 1.25 ).

Step 5:
Verification of the original values given by substituting for z in Equation 1: substitute z=16
(√361 – 15z) – (√z+ 209) = - 4 ; (√361-240) – (√225) = (√121) - (√225) = 11 – 15 = -4 -- Verified

The equation 2 that we created is also satisfied:
(√361 – 15z) + (√z+ 209) = 4z-38
Substituting z=16, and simplifying, we get

11+15 = 64-38 = 26 - Verified..

This completes the Proof and the Analysis

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