Show all work to solve the equation for x. If a solution is extraneous, be sure to identify it in your final answer.

2 works

-3 is extraneous

Step-by-step explanation:

sqrt(x + 7) - 1 = x                      Add 1 to both sides

sqrt(x + 7) = x + 1                     Square both sides

(x + 7) = (x + 1 )^2

x + 7 = x^2 + 2x + 1                Subtract x from both sides

0 = x^2 + x + 1 -7                    

0 = x^2 + x - 6                        This will factor

(x + 3)(x - 2)  = 0                      Find the values for x

x + 3 = 0

x = - 3

x - 2 = 0

x = 2

Now you must check this to see if both values work

sqrt(-3 + 7)  - 1 = -3

sqrt(4) - 1 = -3

2 - 1 =? - 3

1 does not equal - 3. Therefore - 3 is extraneous. Try x = 2

sqrt(2 + 7) - 1 = 2

sqrt(9) - 1 = 2

3 - 1 = 2

2 = 2

tell you what. i'll find the length of the medians and you can fill in the blanks where they belong.

br

b = (2a,0)

r = (a/2,b/2) remember these things are medians. they go to the 1/2 way point of the line opposite the vertex they face.

br^2 =   (2a - a/2)^2 + (0 - b/2)^2

br^2 = (3/2 a) ^2 + b^2 / 4

br^2 = 9/4 a^2 + b^2 / 4   we need to find some relationship between a and b.

let's try ab = bc  

ab = 2a

bc = (2a - a)^2 + (b - 0)^2

bc = sqrt(a^2 + b^2)

ab = bc

2a = sqrt(a^2 + b^2) square both sides.

4a^2 = a^2 + b^2 subtract a^2 from both sides.

sqrt(3a^2) = sqrt(b^2)

sqrt(3)a = b

let's put b into br^2

br^2 = 9/4 a^2 + 3b^2 / 4

br^2 = 12 a^2 / 4

br^2 = 3a^2 

br = sqrt(3) * a

cp

c = (a,b) ; p = (a,0) this is another application of the distance formula, and it is a good one.

cp^2 = (a - a)^2 + (b - 0)^2

cp^2 = 0 + b^2

cp^2 = b^2 which you can see from the diagram. 

cp = sqrt(b^2)

cp = b but b = sqrt(3) * a 

cp = sqrt(3)*a

aq

a = (0,0)

q = (3/2) a, b/2

aq^2 = (3a/2 -0)^2 + (b/2 - 0)^2

aq^2 = 9a^2/4 + b^2/4

but b = sqrt(3) * a 

aq^2 = 9a^2 / 4 + (sqrt(3)*a)^2 /4

aq^2 = 9 a^2/ 4 + 3 a^2 / 4

aq^2 = 12 a^2/4

aq^2 = 3 a^2

aq = (sqrt  3) * a which agrees with the other two.