The Sum Of Two Numbers Is 10. The Difference Of Their Squares Is 4 Less Than Their Product. What Are The Two Numbers?
I'm so confused... just about pulling my hair out if I were not already balding. Haha. If somebody could please help me figure this out and point me to some links for additional help. GOING CRAZY. I know how to do the quadratic equation but I can't figure out how to put it together. Thanks ahead of time for the help.
ARgh. I've been working on this most of the day and I pretty much worked around it. Doing all of the equations that were not in word form. Can someone please help me out with a good link that has indepth explanation of the verbiage used in this. My math book is not very educational in outlining how to differentiate. Thanks.
urbanbub
Let the two numbers be "a" & "b"
Now sum of a and b = 10 (First sentence) = a+b
Difference of their squares = a^2 - b^2
But it is given that this difference is 4 less than their product
Hence
a^2 -b^2 = ab - 4
so you got 2 equations
-----------------------------------------------------
Next Part is solving for a and b
First of all do you remember these formulas
x^2 - y^2 = (x+y)(x-y)...........(0)
(x+y)^2 = x^2 + y^2 - 2xy..................(1)
(x-y)^2 = x^2 +y^2 -2xy ....................(2)
Using 1 and 2
(x+y)^2 = (x-y)^2 +4xy
(x+y)^2 - (x-y)^2 = 4xy..............4
--------------------------------------------------------------
Using (0) we get
a^2 - b^2 = (a+b)(a-b)
But
a^2 - b^2 = ab-4
So
(a+b)(a-b) = ab-4
-----------------------------------------------------
Using 4 we get
(a+b)(a-b) = ((a+b)^2 -(a-b)^2)/4 -4
Put (a-b) = t
&
a+b = 10
10t = (100-t^2)/4 - 4
40t = 84 - t^2
t^2 + 40t -84 = 0
t^2 + 42t -2t -84 = 0
t(t+42) - 2(t+42)= 0
t = 2
or t = -42
Since the numbers should satisfy both equations so t = 2
Hence
a+b = 10 , a-b = 2
So a = 6 & b = 4
----------------------------------------------
Read this from algebra-help.com though I will say its of not much use here
Adarsh
Okay; then let's do it with one variable in one equation!
i) Pick a variable to stand for one of the numbers. Let's say we pick "x".
ii) Create an expression, in terms of the variable in (i), which represents the math for the English expression "their sum is ten". (Hint: If the total is 10, and you've already accounted for x, then the other number, being what's left, must be 10 - x.)
iii) Create expressions, in terms of the variable in (i) and the expression in (ii), for the squares of the two values.
iv) Create an expression, in terms of (iii), for "the difference of the squares".
v) Create an expression, in terms of (i) and (ii), for "their product".
vi) Create an expression, in terms of (v), for "their product, less four".
vii) Create an equation, in terms of (iv) and (v), for "(the difference of the squares) is (their product, less four)".
Then solve the quadratic equation! (You can use factoring, but the Quadratic Formula might be easier in this case.)