Question 483542: i have no idea where to go on this problem please help
"joe said, if five times my age in 8 years is subtracted from the square of my present age, the result is 86." find joe's age
Found 2 solutions by Learners04, bucky:
Answer by Learners04(5)   (Show Source):
You can put this solution on YOUR website! "joe said, if five times my age in 8 years is subtracted from the square of my present age, the result is 86." find joe's age
let joe's present age be x
so joe's age 8 yrs to this the present time will be x+8,
now that means 5* joe's age in 8yrs is 5(x+8)
the square of his present age is x^2
therefore we can hereby write that
X^2-5(x+8)=86: expanding the bracket we'll have
X^2-5x-40=86
x^2-5x-40-86=0
x^2-5x-126=0
can you predict the factors of this quadratic equation?
try
it"s 14 and 9
x^2-14x+9x-126=0
x(x-14)+9(x-14)=0
(x+9)(x-14)=0
either x+9=0 or x-14=0
the first option gives x=-9 while the second gives x as 14
since joe's age cannot be negative, joe's present age is 14yrs
hope you understand the tricks
look into your maths book or this site and solve related problems to perfect your skills.
Answer by bucky(2189)  (Show Source):
You can put this solution on YOUR website! To solve problems such as this, you have to pick them apart ... work by word.
.
First, the problem asks you to find Joe's age (meaning Joe's present or current age). So we have an unknown ... Joe's present age. Let's call this unknown "J" for short.
.
Joe says "if five times my age in 8 years ..." Let's stop here for a bit. In 8 years from now, Joe will be 8 years older than he is now. So in 8 years he will be 8 years older than J, his present age. In math terms, 8 years from now Joe will be J+8 years old. Therefore, 5 times his age in 8 years will be 5 times the quantity J+8. In math terms that can be expressed as 5*(J+8).
.
Next Joe says that 5 times his age in 8 years [which we have expressed as 5*(J+8) is subtracted from his present age squared ... Let's work on this. His present age (which we called J) squared is J-squared or J^2 for short. From this we need to subtract 5*(J+8).
.
So far we have J^2 - 5*(J+8) as the math way of writing 5 times his age 8 years from now subtracted from his present age squared.
.
Now Joe states that this expression must equal 86. So we can now write:
.
J^2 - 5*(J+8) = 86
.
There we have the problem set up as a mathematical equation (a quadratic or second degree equation because of the exponent 2 on the J term). Now all we have to do is to apply some math rules to solve for J.
.
Let's continue by doing the distributed multiplication of -5 times the quantity J+8. We do that by multiplying -5 times each of the terms in the parentheses. When we do that multiplication, the equation becomes:
.
J^2 - 5J - 40 = 86
.
Next, in solving quadratic equations, we generally try to get our equation into the standard form:
.
aJ^2 + bJ + c = 0
.
We sort of have some of this form already. If a = 1, then the first term of our equation is just the J^2 term. Next if b = -5, then we have the bJ we have the correct bJ form. But we have a problem. The term c is just a constant and we also don't have a zero on the right side so that we are in the standard form. We can get a zero on the right side by subtracting 86 from both sides as follows:
.
J^2 - 5J - 40 - 86 = 86 - 86
.
On the left side, the -40 and -86 combine to -126 and on the right side the 86 and -86 total zero. So in standard form our equation becomes:
.
J^2 - 5J - 126 = 0
.
and comparing this to the standard form we see that a = 1, b = -5, and c = -126. Now we can solve the problem.
.
Solving a quadratic equation can be done graphically, by factoring, by completing the square, or by applying the quadratic formula which is just another way of completing the square. The quadratic formula is a way that always works. It is usually more work than factoring, but it can always be counted on to work, even in cases where the equation does not factor. Let's factor our equation first.
.
To make it easier, if you play around a bit you will find that the left side of our equation factors as shown below and it has to equal zero:
.
(J + 9)*(J - 14) = 0
.
Notice that this equation will be true if either of the factors equals zero because zero times anything equals the zero on the right side. So our answers for J can be found by setting each factor equal to zero as follows:
.
J + 9 = 0 and J - 14 = 0
.
By adding - 9 to both sides for the first factor we get J = -9. That means that J, Joe's present age, can be -9. A negative age doesn't make sense so we ignore this answer.
.
By adding +14 to both sides for the second factor we get J = +14. This is the answer we are looking for. Joe's present age (J) is 14 years. In 8 years he will be 22 and 5 times that is 110. If we square his present age (14 squared) we get 196 and then subtract 110 (which is 5 times his age 8 years from now) we get 86 as the answer, just as the problem says it should be. So the answer checks ... Joe's present age is 14.
.
We also could have solved this using the quadratic formula as follows:
.
.
From the equation we got into the standard form we already saw that a = 1, b = -5, and c = -126. If we substitute these values into the quadratic formula we get:
.
.
This simplifies to:
.
.
And the term in the radical 25+504 equals 529. So this further simplifies to:
.
.
The square root of 529 is 23 so we get:
.
.
So the two answers are:
.
 which results in J = -9
.
or
.
 which results in J = +14
.
These are the same two answers as we got by factoring which means that Joe's current age is +14.
.
Hope this helps you to understand how to analyze a problem so that you can set up an equation that will let you solve it. And hopefully you learned a little bit about solving quadratic equations also.
.
|
|
|
