SOLUTION: Mary is 12 years older than her sister, Martha. The product of their ages is 540. How old is each?

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Question 229581: Mary is 12 years older than her sister, Martha. The product of their ages is 540. How old is each?
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
Mary is 12 years older than her sister, Martha. The product of their ages is 540. How old is each?

Step 1. Let x be the age of Martha.

Step 2. Let x+12 be the age of Mary.

Step 3. Then x%28x%2B12%29=x%5E2%2B12x=540 since the product of their ages is 540.

Step 4. We can form a quadratic equation by subtracting 540 from both sides of the equation

x%5E2%2B12x-540=cross%28540%29-cross%28540%29

x%5E2%2B12x-540=0

Step 4. To solve, use the quadratic formula given as

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

where a=1, b=12 and c=-540.

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B12x%2B-540+=+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%2A1%2A-540=2304.

Discriminant d=2304 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-12%2B-sqrt%28+2304+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2812%29%2Bsqrt%28+2304+%29%29%2F2%5C1+=+18
x%5B2%5D+=+%28-%2812%29-sqrt%28+2304+%29%29%2F2%5C1+=+-30

Quadratic expression 1x%5E2%2B12x%2B-540 can be factored:
1x%5E2%2B12x%2B-540+=+1%28x-18%29%2A%28x--30%29
Again, the answer is: 18, -30. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B12%2Ax%2B-540+%29



Step 5. Selecting the positive answer x=18 then x+12=30 and note the product of 18 and 30 is 540.

Step 6. ANSWER: Martha is 18 years old and Mary is 30 years old.

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.

Good luck in your studies!

Respectfully,
Dr J
S