SOLUTION: There is a rectangle with 2 diagonals. The diagonals are RT and QS with intersection C. The problem gives you RT as 3xsquared and QC as 5x+4. Thye want you to find the value of x

Algebra ->  Rectangles -> SOLUTION: There is a rectangle with 2 diagonals. The diagonals are RT and QS with intersection C. The problem gives you RT as 3xsquared and QC as 5x+4. Thye want you to find the value of x      Log On


   

Question 171781This question is from textbook Geometry: Integration, Application and Connections
: There is a rectangle with 2 diagonals. The diagonals are RT and QS with intersection C. The problem gives you RT as 3xsquared and QC as 5x+4. Thye want you to find the value of x. This question is from textbook Geometry: Integration, Application and Connections

Found 2 solutions by nerdybill, Earlsdon:
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
Since the diagonals are equal, this means RT = QC
.
3x^2 = 5x+4
3x^2 - 5x - 4 = 0
.
Since you can't factor you must resort to the quadratic equation. Doing so will yield the following solutions:
x = {2.257, -0.591}
.
You can toss out the negative solution leaving:
x = 2.257
.
Details of quadratic follows:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B-5x%2B-4+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-5%29%5E2-4%2A3%2A-4=73.

Discriminant d=73 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--5%2B-sqrt%28+73+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-5%29%2Bsqrt%28+73+%29%29%2F2%5C3+=+2.25733395755292
x%5B2%5D+=+%28-%28-5%29-sqrt%28+73+%29%29%2F2%5C3+=+-0.590667290886255

Quadratic expression 3x%5E2%2B-5x%2B-4 can be factored:
3x%5E2%2B-5x%2B-4+=+3%28x-2.25733395755292%29%2A%28x--0.590667290886255%29
Again, the answer is: 2.25733395755292, -0.590667290886255. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-5%2Ax%2B-4+%29

(3x+2)(x-2) = 0

Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
My appologies to Nerdy Bill but the solution is a bit off.
From the description of the rectangle and its diagonals, the one diagonal is RT+=+3x%5E2 but the other diagonal QS is really given as QC+=+5x%2B4 and this is but half of the complete diagonal QS, so the equation becomes:
RT+=+2QC or
3x%5E2+=+2%285x%2B4%29 Rewriting this it becomes:
3x%5E2-10x-8+=+0 ...and this is factorable to:
%283x%2B2%29%28x-4%29+=+0 and so...
x+=+-2%2F3 or x+=+4 and, as you pointed out, you can discard the negative quantity as we are talking about lengths, so...
x = 4
Check:
3x%5E2+=+3%284%29%5E2 = 3%2816%29+=+48
2%285x%2B4%29+=+2%2820%2B4%29 = 2%2824%29+=+48