SOLUTION: The height of a triangle is 5 centimeters longer than three times its base. Find the base of the triangle if its area is 6 square centimeters

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Question 212048This question is from textbook
: The height of a triangle is 5 centimeters longer than three times its base. Find the base of the triangle if its area is 6 square centimeters This question is from textbook

Answer by drj(1380) About Me  (Show Source):
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The height of a triangle is 5 centimeters longer than three times its base. Find the base of the triangle if its area is 6 square centimeters

Step 1. Area of Triangle = A+=+xh%2F2+=6

where x=base, h=height and A=area of is one-half times base times height

Step 2. h=3x%2B5 (The height of a triangle is 5 centimeters longer than three times its base)

Step 3. Substitute height in Step 2 into equation in Step 1.

A=xh%2F2=x%283%2Bx5%29%2F2
%283x%5E2%2B5x%29%2F2=6

Step 4. Multiply 2 in both sides of last equation in Step 3 to get rid of denominator.

2%283x%5E2%2B5x%29%2F2=2%2A6
3x%5E2%2B5x=12
Simplify by subtracting 12 from both sides of equation yields

3x%5E2%2B5x-12=0

Step 5. The last equation of Step 4 is simply a quadratic equation. Then use the quadratic formula:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=3, b=5, and c=-12

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B5x%2B-12+=+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=%285%29%5E2-4%2A3%2A-12=169.

Discriminant d=169 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-5%2B-sqrt%28+169+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%285%29%2Bsqrt%28+169+%29%29%2F2%5C3+=+1.33333333333333
x%5B2%5D+=+%28-%285%29-sqrt%28+169+%29%29%2F2%5C3+=+-3

Quadratic expression 3x%5E2%2B5x%2B-12 can be factored:
3x%5E2%2B5x%2B-12+=+3%28x-1.33333333333333%29%2A%28x--3%29
Again, the answer is: 1.33333333333333, -3. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B5%2Ax%2B-12+%29



Step 6. Pick the positive number in the quadratic equation. In this case, we have the base x= 1.3333=4/3.

height is then h=3x%2B5=3%284%2F3%29%2B5=9 centimeters
Area is then %281%2F2%29%284%2F3%29%289%29=6 square centimeters. This is the same answer to our given answer. So it check out.

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