SOLUTION: the difference of two whole numbersis 2. if two times the squareof the smaller number added too three times the squareof the larger numbers equal 140, find the numbers

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Question 216488: the difference of two whole numbersis 2. if two times the squareof the smaller number added too three times the squareof the larger numbers equal 140, find the numbers
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
The difference of two whole numbers is 2. if two times the square of the smaller number added too three times the square of the larger numbers equal 140, find the numbers.

Step 1. Let x be one number and x+2 be the other number.

Step 2. Let 2x%5E2+ be two times the square of the smaller number.

Step 3. Let 3*(x+2)^2 be three times the square of the larger number.

Step 4. The, 2x%5E2%2B3%28x%2B2%29%5E2=140 since two times the square of the smaller number added too three times the square of the larger numbers equal 140.

Step 5. Solving yields the following steps:

2x%5E2%2B3%28x%2B2%29%5E2=140

2x%5E2%2B3%28x%5E2%2B4x%2B4%29=140

2x%5E2%2B3x%5E2%2B12x%2B12=140

5x%5E2%2B12x%2B12=140

Subtract 140 from both sides of the equation to get a quadratic equation.

5x%5E2%2B12x%2B12-140=140-140

5x%5E2%2B12x-128=0

Step 6. To solve, we can use the quadratic equation given as

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

where a=5, b=12, and c=-128

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 5x%5E2%2B12x%2B-128+=+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=%2812%29%5E2-4%2A5%2A-128=2704.

Discriminant d=2704 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-12%2B-sqrt%28+2704+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2812%29%2Bsqrt%28+2704+%29%29%2F2%5C5+=+4
x%5B2%5D+=+%28-%2812%29-sqrt%28+2704+%29%29%2F2%5C5+=+-6.4

Quadratic expression 5x%5E2%2B12x%2B-128 can be factored:
5x%5E2%2B12x%2B-128+=+5%28x-4%29%2A%28x--6.4%29
Again, the answer is: 4, -6.4. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+5%2Ax%5E2%2B12%2Ax%2B-128+%29



Select the positive number since we want whole numbers gives x=4 and x+2=6

Check...2x%5E2%2B3%28x%2B2%29%5E2=2%2A4%5E2%2B3%2A6%5E2=32%2B108=140 which is a true statement.

Step 7. ANSWER: The numbers are 4 and 6.

I hope the above steps were helpful.

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Good luck in your studies!

Respectfully,
Dr J