SOLUTION: A mother is three times as old as her daughter. Four years ago the product of their ages was 256. Find their current ages. Would the equation be (3x-4)(x-4)=256?

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Question 852410: A mother is three times as old as her daughter. Four years ago the product of their ages was 256. Find their current ages. Would the equation be (3x-4)(x-4)=256?
Answer by pmesler(52) About Me  (Show Source):
You can put this solution on YOUR website!
Let the daughter's age = x
Let the mother's age = 3x

The product of their ages is 256. In other words
(3x-4)(x-4) = 256
Use FOIL to simplify.
3x^2-16x+16 = 256
Subtract 256 from both sides to make this a quadratic equation.

3x^2-16x-240 = 0
Solve for x using the quadratic equation.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B-16x%2B-240+=+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=%28-16%29%5E2-4%2A3%2A-240=3136.

Discriminant d=3136 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--16%2B-sqrt%28+3136+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-16%29%2Bsqrt%28+3136+%29%29%2F2%5C3+=+12
x%5B2%5D+=+%28-%28-16%29-sqrt%28+3136+%29%29%2F2%5C3+=+-6.66666666666667

Quadratic expression 3x%5E2%2B-16x%2B-240 can be factored:
3x%5E2%2B-16x%2B-240+=+3%28x-12%29%2A%28x--6.66666666666667%29
Again, the answer is: 12, -6.66666666666667. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-16%2Ax%2B-240+%29




From this you can see that x = (-6.66,0) and x = (12,0). Obviously we can disregard x = -6.66 since you can't have a negative age. That means that the daughter is 12 years old and her mother is 36 years old.

Let's plug in these values of x to see if the original equation checks out.
(3(12)-4)(12-4) = 256
32 * 8 = 256
It checks out so those are the correct ages.