SOLUTION: please help me solve this problem. the area of a triangle is 13.5 meters. Find the base and height of the retangle if its height is 6 meters greaster than it's base. Use an eq

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Question 166492: please help me solve this problem.
the area of a triangle is 13.5 meters. Find the base and height of the retangle
if its height is 6 meters greaster than it's base. Use an equation and the formula area of a triangle =0.5(base)(height).
Found 2 solutions by nerdybill, gonzo:
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
the area of a triangle is 13.5 meters. Find the base and height of the retangle
if its height is 6 meters greater than it's base. Use an equation and the formula area of a triangle =0.5(base)(height).
.
Let b = length of the base
then because "its height is 6 meters greater than it's base"
b+6 = height
.
Plug the above into:
area of a triangle =0.5(base)(height)
13.6 =0.5(b)(b+6)
27.2 =(b)(b+6)
27.2 = b^2 + 6b
0 = b^2 + 6b - 27.2
.
Using the quadratic equation to solve we get:
b = {3.01664358259653, -9.01664358259653}
Throw out the neg solution.
b = 3 meters (base)
b+6 = 9 meters (height)
.
Quadratic solved here:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ab%5E2%2Bbb%2Bc=0 (in our case 1b%5E2%2B6b%2B-27.2+=+0) has the following solutons:

b%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=%286%29%5E2-4%2A1%2A-27.2=144.8.

Discriminant d=144.8 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-6%2B-sqrt%28+144.8+%29%29%2F2%5Ca.

b%5B1%5D+=+%28-%286%29%2Bsqrt%28+144.8+%29%29%2F2%5C1+=+3.01664358259653
b%5B2%5D+=+%28-%286%29-sqrt%28+144.8+%29%29%2F2%5C1+=+-9.01664358259653

Quadratic expression 1b%5E2%2B6b%2B-27.2 can be factored:
1b%5E2%2B6b%2B-27.2+=+1%28b-3.01664358259653%29%2A%28b--9.01664358259653%29
Again, the answer is: 3.01664358259653, -9.01664358259653. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B6%2Ax%2B-27.2+%29

Answer by gonzo(654) About Me  (Show Source):
You can put this solution on YOUR website!
A = (b*h)/2 (equation 1)
where A = area
b = base
h = height
-----
h = b+6 (given)
A = 13.5 (given)
equation 1 becomes:
A = [b*(b+6)]/2 = 13.5
which becomes:
[b*(b+6)]/2 = 13.5 (equation 2)
-----
multiply both sides of equation 2 by 2:
b*(b+6) = 27
simplify:
b^2 + 6*b = 27
complete the squares:
(b+3)^2 = 27 + 9 (explanation for this down below after the answer)
simplify:
(b+3)^2 = 36
take square root of both sides:
b+3 = +/- 6
subtract 3 from both sides:
b = +/- 6 - 3
b becomes either:
6-3 = 3
or:
-6-3 = -9
-----
since b can't be negative, answer has to be:
b = 3
-----
since h = b+6, then:
h = 9
-----
you have:
b = 3
h = 9
A = 13.5 (given)
-----
substitute in equation 1:
A = (b*h)/2 (equation 1)
A = (3*9)/2 = 13.5
A = 27/2 = 13.5
A = 13.5 = 13.5
-----
values for b and h prove out.
equation is good.
b = 3
h = 9
-----
explanation for completing the squares is shown below:
also shown below is the fact that you could have also solved this using the quadratic formula.
-----
in order to complete the squares of your equation:
b^2 + 6*b)
i took half of the 6 and turned the equation into:
(b+3)^2
if you multiply this out, you will get:
b^2 + 6*b + 9
there's an extra 9 in there.
in order to keep the equations in balance i had to add 9 to the other side of the equation.
that's why the equation went from:
b^2 + 6*b = 27
to:
(b+3)^2 = 27 + 9
the additional 9 on the right hand side of the equation kept them in balance because i added 9 to the left hand side by completing the squares.
-----
use of the quadratic formula instead.
-----
rather than completing the squares, you can also solve this equation using the quadratic formula.
the equation to solve was:
b^2 + 6*b = 27
subtract 27 from both sides of the equation:
b^2 + 6*b - 27 = 0
let x = b (this is necessary because general form of quadratic equation and quadratic formula use b for another purpose)
-----
formula becomes:
x^2 + 6*x - 27 = 0
general form of quadratic equation is:
a*x^2 + b*x + c = 0
the quadratic formula can be used to solve this quadratic equation.
general form of quadratic formula is: x = %28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29
a = 1
b = 6
c = -27
-----
b^2 = 36
4*a*c = -108
2*a = 2
-----
quadratic formula becomes
x = %28-6+%2B-+sqrt%2836-%28-108%29%29%29%2F2
this becomes:
x = (-6 +/- 12)/2
which becomes
x = 6/2 = 3
or:
x = -18/2 = -9
-----
the answers are the same.
you could solve by completing the square or you could solve by quadratic formula, whichever is easiest for you.
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