Question 1052497
let the mother's age be x.
let the daughter's age be y.


the sum of their ages is 29.


this mean that x + y = 29.


4 years ago, the product of their ages was 96.


the mother's age 4 years ago was x - 4.


the daughter's age 4 years ago was y - 4.


the product of their ages 4 years ago was 96.


this mean that (x-4) * (y-4) = 96.


in the equation of x + y = 29, solve for y to get:


y = 29-x.


in the equation of (x-4) * (y-4) = 96, replace y with 29-x to get:


(x-4) * (29-x-4) = 96.


simplify to get:


(x-4) * (25-x) = 96.


multiply the factors together to get:


25x - x^2 - 100 + 4x = 96


combine like terms to get:


29x - x^2 - 100 = 96


subtract 96 from both sides of the equation to get:


29x - x^2 - 196 = 0


rearrange the terms in descending order of degree to get:


-x^2 + 29x - 196 = 0


that's your quadratic equation in standard form.


unfortunately, it doesn't factor cleanly, but it does factor and the answer makes sense, even though it doesn't.


i used a quadratic equation calculator to find the solution.


the solution to this quadratic equation of -x^2 + 29x - 196 = 0 are:


x = 18.274917217635 or x = 10.725082782365


since x + y = 29, then:


when x = 18.274917217635, y = 10.725082782365


and when x = 10.725082782365, y = 18.274917217635


since x is the age of the mother, then x has to be greater than y, so you get:


age of the mother is 18.274917217635


age of the daughter is 10.725082782365


the sum of their ages is 29.


if you subtract 4 from each of their ages and then multiply those ages together, you get 96.


the solution is 


since the mother has to be older than the daughter, then let x = 18.274917217635 and y = 10.725082782365.


not very realistic, but that's how the numbers come out.


the mother would have had to have the daughter when she was 8.


not likely.


add the ages together and you get 29.


multiply 14.274917217635 * 6.725082782365 and you get 96.


the requirements of the problem are satisfied.


therefore the solution looks good.


that's the part that makes sense.


the part that doesn't make sense is the mother is only 8 years older than the daughter.


that's not very likely.


in fact, it may even be impossible.


you did not need to go this far however.


all they wanted was the quadratic equation.


the solution you are looking for is:


the standard form of the quadratic equation is -x^2 + 29x - 196 = 0


NOW TO YOUR SECOND PROBLEM.


the difference of two natural numbers is 3 and difference of their reciprocals is 3/28 forms a quadratic equation. 


let x equal the first number and y equal the second number.


these are natural numbers so they have to be positive integers i believe.
they can sometimes also include 0, depending on who's defining.
they are definitely not negative.


if x is the larger number, then you get x - y = 3.


the difference of their reciprocals is 3/28.


this means that 1/y - 1/x = 3/28


note that if x > y, then 1/x < 1/y which makes 1/y - 1/x positive.


you can use any positive numbers for x and y to confirm this is true.


so we have two equations.


x - y = 3


1/y - 1/x = 3/28


if we multiply both sides of the second equation by xy, we get:


x - y = 3xy/28


if we multiply both sides of this equation by 28, we get:


28x - 28y = 3xy


from the equation of x - y = 3, we can solve for x to get:


x = y+3


in the equation of 28x - 28y = 3xy, replace x with y+3 to get:


28(y+3) - 28y = 3(y+3)y


simplify to get:


28y + 84 - 28y = 3y^2 + 9y 


combine like terms to get 84 = 3y^2 + 9y


subtract 84 from both sides of the equation to get 0 = 3y^2 + 9y -84


flip sides to get 3y^2 + 9y - 84 = 0


that's your quadratic equation in standard form.


factor this equation to get (3y-12) * (y+7) = 0


solve for y to get y = 4 or y = -7


y has to be a positive integer.


therefore, y has to be equal to 4.


let's see if that's correct.


y is the smaller number.


if x - y = 3, then x - 4 = 3 which makes x = 7.


you have x = 7 and y = 4


7 - 4 = 3


1/4 - 1/7 should be equal to 3/28.


common denominator is 28.


you get 7/28 - 4/28 = 3/28.


it does equal to 3/28, so the solution looks good.l


the solution is x = 7 and y = 4.