SOLUTION: Factor and quadratic formulas. I must find the exact solution. If the solutions are irrational then I must find the approximate solution. PROBLEM:The width of a rectangle gate

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: Factor and quadratic formulas. I must find the exact solution. If the solutions are irrational then I must find the approximate solution. PROBLEM:The width of a rectangle gate       Log On


   

Question 982262: Factor and quadratic formulas.
I must find the exact solution. If the solutions are irrational then I must find the approximate solution.
PROBLEM:The width of a rectangle gate is 2 meters (m) larger than the height.the diagonal brace measures the root of 6m. Find the width and height.
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
.
H=Height; W=Width=H%2B2m; D=diagonal=sqrt%286%29m
.
The height and width are two legs of a right triangle with diagonal as hypotenuse.
.
H%5E2%2BW%5E2=D%5E2 Substitute for W.
H%5E2%2B%28H%2B2m%29%5E2=%28sqrt%286%29m%29%5E2
H%5E2%2BH%5E2%2B4Hm%2B4m%5E2=6m%5E2
2H%5E2%2B4Hm-2m%5E2=0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aH%5E2%2BbH%2Bc=0 (in our case 2H%5E2%2B4H%2B-2+=+0) has the following solutons:

H%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=%284%29%5E2-4%2A2%2A-2=32.

Discriminant d=32 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-4%2B-sqrt%28+32+%29%29%2F2%5Ca.

H%5B1%5D+=+%28-%284%29%2Bsqrt%28+32+%29%29%2F2%5C2+=+0.414213562373095
H%5B2%5D+=+%28-%284%29-sqrt%28+32+%29%29%2F2%5C2+=+-2.41421356237309

Quadratic expression 2H%5E2%2B4H%2B-2 can be factored:
2H%5E2%2B4H%2B-2+=+2%28H-0.414213562373095%29%2A%28H--2.41421356237309%29
Again, the answer is: 0.414213562373095, -2.41421356237309. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B4%2Ax%2B-2+%29

H=0.4142 meters (approx)