Solve sqrt (x + 73) - sqrt (x + 58) = 1?

This problem can easily be solved by plugging in the answers provided. You'll quickly find that the answer is (a) -9.

sqrt(73-9) - sqrt(58-9) = sqrt(64) - sqrt(49) = 1

The others don't work.

If you had to solve this algebraically, start by squaring both sides:

(x+73) - 2 sqrt[(x+73)(x+58)] + (x+58) = 1
2x + 130 = 2 sqrt[(x+73)(x+58)]
x+65 = sqrt[(x+73)(x+58)]

Square both sides AGAIN, and
(x+65)^2 = (x+73)(x+58)

x^2 + 130x + 4225 = x^2 + 131x + 4234
x = 4225-4234 = -9

Multiply by the conjugate (the conjugate is sqrt(x+73) + sqrt(x+58) by changing the - to a +):

(x+73) - (x + 58) = sqrt(x+73) + sqrt(x+58)

15 = sqrt(x+73) + sqrt(x+58)

Add this to the original equation:

[sqrt(x+73) + sqrt(x+58)] + [sqrt(x+73) - sqrt(x+58)] = 15 + 1

2*sqrt(x+73) = 16

and solving this equation gives:

sqrt(x+73) = 8
x+73 = 64
x = -9

Try them all, and see which one fits.

sqrt(64) - sqrt(49) = 8 - 7 = 1

x+sqrt(x)=1 If I multiply by x, I get x^2+x=x this is wrong If you multiply by x (I am not saying it is the best thing to do) the result is x(x+√x=1) x^2 + x √x = x this is not very useful, but at least is correct :-) the best you can do is bring the x into the right side √x = 1 - x (*) and square both sides (this is at risk, because it can 'create' roots that are not solution of the original equation) x = (1 - x)^2 x = 1 - 2x + x^2 x^2 - 3x + 1 = 0 (**) x = (3 ± √8)/2 x₁ = (3 - √8)/2 ≈ 0.38 x₂ = (3 + √8)/2 ≈ 2.62 the two roots are solution of (**) but it is not sure they are also solution of (*) indeed, looking carefully equation (*) we notice that as √x is positive for any x>0, right side 1 - x is positive only if x < 1 therefore x₂ ≈ 2.62 is not a solution of (*) Let's check if x₁ ≈ 0.38 is solution √0.38 ≈ 1 - 0.38 it's true the unique solution of the given equation is x = (3 - √8)/2

√(x + 73) - √(x + 58) = 1
√(x + 73) = 1 + √(x + 58)
x + 73 = [1 + √(x + 58)]^2
x + 73 = 1 + 2√(x + 58) + (x + 58)
x + 73 = x + 58 + 1 + 2√(x + 58)
x - x + 73 - 59 = 2√(x + 58)
14/2 = √(x + 58)
x + 58 = 7^2
x = 49 - 58
x = -9
(answer a)

sqrt (x + 73) - sqrt (x + 58) = 1
square both sides

(x+73) - 2 sqrt(x+73) sqrt(x+58) + (x+58) = 1
(using (a-b)^2 = a^2-2ab+b^2 formula)

x+73 + x + 58 = 1+ 2 sqrt(x+73) sqrt(x+58)
2x+131 = 1+ 2 sqrt(x+73) sqrt(x+58)
2x+130 = 2 sqrt(x+73)sqrt(x+58)
square both sides again
(2x+130)^2 = 4 (x+73)(x+58)

4x^2+520x+169,00 = 4(x^2+73x+58x+4234)
4x^2+520x+169,00 = 4x^2+524x+169,36
-4x = 36
x=-9

Multiple choice makes things easy :) As shown, x = -9 is a solution, but x = 8 is another solution. Try it and you'll see.

sqrt ((x) + 73) - sqrt ((x) + 58) = 1;
sqrt ((-9) + 73) - sqrt ((-9) + 58) = 1;
sqrt (64) - sqrt (49) = 1;
8 - 7 = 1;
1 = 1;
So x = -9(a is the ans)

Answer is a -9//

a is correct.

73-9 =64, sqrt 64 = 8
58-9 = 49 sqrt 49 = 7