Question

Find the solution of 3 times the square root of the quantity of x plus 6 equals negative 12, and determine if it is an extraneous solution.

Answer 1

I'll do the last one, since it's probably most difficult because of x on both sides. Hopefully, this will help you do at least some of the other problems. The strategy for solving square root equations is to isolate the radical, square both sides to get rid of the radical, solve the resulting equation, and check for extraneous (invalid) solutions by plugging your solutions back into the original equation.

I will use sqrt(expression) for the square root of an expression.

sqrt(x + 2) + 4 = x
sqrt(x + 2) = x - 4, from subtracting 4 from both sides to isolate radical
[sqrt(x + 2)]^2 = (x - 4)^2, from squaring both sides (this can create extraneous solutions)
x + 2 = x^2 - 8x + 16, from simplifying each side
0 = x^2 - 9x + 14, from adding -2 - x to both sides to get 0 on one side
0 = (x - 7)(x - 2), from factoring
x - 7 = 0 or x - 2 = 0, from using the zero factor property
x = 7 or x = 2

Plug in x = 7 to sqrt(x + 2) + 4 = x:
sqrt(7 + 2) + 4 =? 7
sqrt(9) + 4 =? 7
3 + 4 = 7, so x = 7 works.

Plug in x = 2 to sqrt(x + 2) + 4 = x:
sqrt(2 + 2) + 4 =? 2
sqrt(4) + 4 =? 2
2 + 4 is unequal to 2, so x = 2 fails and is thus extraneous.

The only valid solution is x = 7.

Lord bless you today!

Answer 2

dang