Math - I need help with solving radical equations and geometric applications?

Question

Solve.
____
1.) 3 \/ x+2 - 4 = 3

Solve
____ ____
2.) \/ x+3 - \/2x-8 = 1



3.) a 25ft latter is leaning against a building. If the foot of the ladder is 15 feet from the base of the building, how far up the side of the building does the ladder reach?


4.) In a 30-60-90, find the length of the hypothenuse if the side opposite the 60deg angle measures 9.

| \
| \
9| \
|____\ <- 60deg

Answer 1

1.) 3 \/ x+2 - 4 = 3
if you mean
3 sqrt( x+2) -4 =3,
then
3 sqrt(x+2) =7
sqrt(x+2) = 7/3
x+2 = 49/9
x=49/9 -2

2.) \/ x+3 - \/2x-8 = 1
sqrt(x+3) -sqrt(2x-8) =1
sqrt(x+3) = sqrt(2x-8) +1
x+3 = 2x-8 +2 sqrt(2x-8) +1
-x + 10 = 2 sqrt(2x-8)
x^2 -20x + 100 = 4 ( 2x-8)
x^2 - 28x +132 =0
'

Answer 2

Since you didn't put your square root quantity into parentheses, and the limits of typing don't put the handy-dandy bar over the top of the quantity, I'm going to guess at what you meant. Check these because if I got that wrong, the whole thing will be wrong:

1. 3(sqrt)(x + 2) - 4 = 3
3 (sqrt)(x + 2) = 7
sqrt(x + 2) = 7/3
x + 2 = +/-(sqrt)(7/3)
x = -2 +/-(sqrt)(7/3)

2. sqrt(x + 3) - sqrt(2x - 8) = 1
(sqrt(x + 3) - sqrt(2x - 8) = 1)^2
(x + 3) -2(sqrt)(2x^2 - 2x - 24) + (2x - 8) = 1
3x - 5 - 2(sqrt)(2x^2 - 2x - 24) = 1
3x - 6 = 2(sqrt)(2x^2 - 2x - 24)
(3x - 6 = 2(sqrt)(2x^2 - 2x - 24))^2
3x - 6 = 4(2x^2 - 2x - 24)
3x - 6 = 8x ^2 -8x -96
8x ^2 -11x -90 = 0

(There's got to be a better way to do this, but I can't think of one.)
Since this doesn't look like something that will easily factor, you can either use the quadratic equation or completing the square to get your answer. I prefer the latter, which goes a little somethin' like this:

Divide out your x^2 coefficient:
8x ^2 -11x -90 = 0
x^2 - 11/8x - 45/4 = 0

Put the constant on the right side of the equation:
x^2 - 11/8x = 45/4

Divide the x coefficient by 2, square it, and add it to both sides of the equation:
x^2 - 11/8x + 121/256 = 45/4 + 121/256

Factor the left side of the equation to (x + a)^2, where a is the number you just squared:
(x - 11/16)^2 = 3001/256

Take the square root of both sides, remembering that this will leave you with both a positive and a negative root:
x - 11/16 = (sqrt)(3001/256)

Move the constant from the left to the right side and simplify:
x = 11/16 +/- (sqrt)(3001/256)
x = 11/16 +/- (sqrt 3001)/16
x = (11 +/- sqrt 3001)/16

3. The ladder, the ground, and the building are forming a right triangle. Since the hypotenuse of the triangle (the ladder) is in a 5:3 ratio to one of the other two sides (the ground), this must be a 3-4-5 right triangle. Thus, the ladder reaches 20 feet up the side of the building.

4. The ratio of sides for a 30-60-90 triangle is, in order of the measurement of the opposite angle, 1:sqrt 3:2. If the side opposite the 60 degree angle is 9, the hypotenuse must be 2/(sqrt 3) times the length of that side:
h = (9)(2)/(sqrt 3)
h = 18/(sqrt 3)
h = 18(sqrt 3)/3
h = 6(sqrt 3)

Answer 3

1.
3√(x+2) = 7
√(x+2) = 7/3 square both sides
x+2 = 49/9
x = 49/9 - 2 = 31/9 = 3 4/9

2.
√(x+3) - √(2x-8) = 1
√(x+3) = 1 + √(2x-8) square both sides
x+3 = 1 + 2√(2x-8) + 2x-8
-x + 10 = 2√(2x-8) square both sides
x² -20x + 100 = 4(2x-8) = 8x - 32
x² - 28x + 132 = 0
(x - 22) (x - 6) = 0
x = 22 or 6

check in original equation:
x=22: √(22+3) - √(44-8) = √25 - √36 = 5 - 6 = -1 != 1
x=6: √(6+3) - √(12-8) = √9 - √4 = 3 - 2 = 1

So only x=6.

3. This is a right triangle. 25² = 15² + x² -> x = 20 (a 3-4-5 triangle in disguise)

4. The longer side of a 30-60-90 is √3 times the shorter side, so the shorter side is 9/√3 = 9√3/3 = 3√3. The hypotenuse is twice the shorter side, or 6√3.

Answer 4

3x^2 - 8 = 21; 3x^2 = 29, x^2 = 29/3, x = ±?29/?3 t^2 = sixty 3; t = ?sixty 3 = ±3?7 9x^2 + 15 = 40; 9x^2 = 25, x^2 = 25/9, x = ±5/3 4x^2 - 8 = 0; 4x^2 = 8, x = ±?2 I figured you may decimalize those in case you needed