SOLUTION: find 2 consecutive integers such that three times the square of the first is equal to seven more than five times the second. what is the answer?

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Question 225261: find 2 consecutive integers such that three times the square of the first is equal to seven more than five times the second.
what is the answer?
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
Find 2 consecutive integers such that three times the square of the first is equal to seven more than five times the second.
What is the answer?

Step 1. Let n be the first integer.

Step 2. Let n%2B1 be the next consecutive integer

Step 3. Let 3n%5E2 be three times the square of the first.

Step 4. Let 7%2B5%28n%2B1%29 be seven more than five times the second.

Step 5. Then 3n%5E2=7%2B5%28n%2B1%29 since three times the square of the first is equal to seven more than five times the second.

Step 6. Solving the equation in Step 5 yields the following steps.

3n%5E2=7%2B5%28n%2B1%29

3n%5E2=7%2B5n%2B5=12%2B5n

Subtract 5n+12 from both sides of the equation

3n%5E2-5n-12=12%2B5n-5n-12

3n%5E2-5n-12=0

Step 7. To solve 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=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 1x%5E2%2B-2x%2B-3+=+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=%28-2%29%5E2-4%2A1%2A-3=16.

Discriminant d=16 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2%2B-sqrt%28+16+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+16+%29%29%2F2%5C1+=+3
x%5B2%5D+=+%28-%28-2%29-sqrt%28+16+%29%29%2F2%5C1+=+-1

Quadratic expression 1x%5E2%2B-2x%2B-3 can be factored:
1x%5E2%2B-2x%2B-3+=+1%28x-3%29%2A%28x--1%29
Again, the answer is: 3, -1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-2%2Ax%2B-3+%29



With n=3 then n+1=4. Check 3n%5E2=7%2B5%28n%2B1%29 if true so 3%2A3%5E2=7%2B5%2A4 or 27=27 which is a true statement.

With n=-1 then n+1=0 Check 3n%5E2=7%2B5%28n%2B1%29 if true so 3%2A%28-1%29%5E2=7%2B5%2A0 or 3=7 which is not a true statement.

Step 8. ANSWER: The numbers are 3 and 4

I hope the above steps were helpful.

For FREE Step-By-Step videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra and for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.

And good luck in your studies!

Respectfully,
Dr J
http://www.FreedomUniversity.TV