how to solve by factoring: Car goes 125 mi; if rate increased by 5mi/hr, then trip...

[allegansveritatem]

Here is problem:

56. A car travels for 125 miles at a uniform speed. If the speed is increased by 5 miles per hour, the trip would take 1 hour less time. What is the car's rate of speed?

Here is what I did with it:

. . . . .\(\displaystyle \dfrac{125}{x}\, -\, 1\, =\, \dfrac{125}{x\, +\, 5}\)

. . . . .\(\displaystyle 125x\, +\, 625\, -\, x^2\, -\, 5x\, =\, 125x\)

. . . . .\(\displaystyle 120x\, +\, 625\, -\, x^2\, =\, 125x\)

. . . . .\(\displaystyle 0\, =\, x^2\, +\, 5x\, -\, 625\)

Now, I am constrained to solve this by factoring--I haven't gotten to the chapters that teach other methods of solving quqadratic equations. My question is: Have I set this up wrong? If not, then, can this equation be factored? I can't seem to find the way. Looks prime to me.

 

[MarkFL]

Here is problem:

56. A car travels for 125 miles at a uniform speed. If the speed is increased by 5 miles per hour, the trip would take 1 hour less time. What is the car's rate of speed?

Here is what I did with it:

. . . . .\(\displaystyle \dfrac{125}{x}\, -\, 1\, =\, \dfrac{125}{x\, +\, 5}\)

. . . . .\(\displaystyle 125x\, +\, 625\, -\, x^2\, -\, 5x\, =\, 125x\)

. . . . .\(\displaystyle 120x\, +\, 625\, -\, x^2\, =\, 125x\)

. . . . .\(\displaystyle 0\, =\, x^2\, +\, 5x\, -\, 625\)

Now, I am constrained to solve this by factoring--I haven't gotten to the chapters that teach other methods of solving quqadratic equations. My question is: Have I set this up wrong? If not, then, can this equation be factored? I can't seem to find the way. Looks prime to me.

Using:

\(\displaystyle d=\overline{v}t\)

I would write:

\(\displaystyle 125=vt\)

\(\displaystyle 125=(v+5)(t-1)\)

This system implies:

\(\displaystyle vt=vt-v+5t-5\)

Arrange as:

\(\displaystyle \frac{v}{5}=t-1\)

Substitute into the second of the first 2 equations:

\(\displaystyle 125=(v+5)\frac{v}{5}\)

Now, this may be arranged as:

\(\displaystyle v^2+5v-625=0\)

This is what you have, so good so far.

This quadratic is not going to factor in the standard way, you will need to either complete the square or use the quadratic formula.

 

[Dr.Peterson]

Here is problem:View attachment 10515

Here is what I did with it:

View attachment 10516

Now, I am constrained to solve this by factoring--I haven't gotten to the chapters that teach other methods of solving quqadratic equations. My question is: Have I set this up wrong? If not, then, can this equation be factored? I can't seem to find the way. Looks prime to me.

As MarkFL said, your equation is correct, and it can't be solved by factoring (over the integers). If this problem is in the book before other methods (such as completing the square) have been taught, then it is out of place. If you just found it somewhere else, then I suppose it's your fault that it isn't one you can solve yet.

The polynomial is indeed prime over the integers (that is, prime in the sense you would have been taught). That can be proved by listing all possible factor pairs of -625, or by more advanced methods such as the discriminant.

 

[allegansveritatem]

Using:

\(\displaystyle d=\overline{v}t\)

I would write:

\(\displaystyle 125=vt\)

\(\displaystyle 125=(v+5)(t-1)\)

This system implies:

\(\displaystyle vt=vt-v+5t-5\)

Arrange as:

\(\displaystyle \frac{v}{5}=t-1\)

Substitute into the second of the first 2 equations:

\(\displaystyle 125=(v+5)\frac{v}{5}\)

Now, this may be arranged as:

\(\displaystyle v^2+5v-625=0\)

This is what you have, so good so far.

This quadratic is not going to factor in the standard way, you will need to either complete the square or use the quadratic formula.

Thanks. I never cease to be amazed at how many ways there are to get to where you wan to go in math. Reminds me of the Japanese Buddhist maxim: There are many ways to get to the top of Mt Fuji

 

[allegansveritatem]

As MarkFL said, your equation is correct, and it can't be solved by factoring (over the integers). If this problem is in the book before other methods (such as completing the square) have been taught, then it is out of place. If you just found it somewhere else, then I suppose it's your fault that it isn't one you can solve yet.

The polynomial is indeed prime over the integers (that is, prime in the sense you would have been taught). That can be proved by listing all possible factor pairs of -625, or by more advanced methods such as the discriminant.

Yes, this problem was one of those at the end of the chapter dealing with solving problems by means of rational expressions. The chapters on completing the square and the quadratic formula come later. I really don't want to fool with that stuff until I come to it. Maybe this problem was put where it was on the supposition that nobody was probably going to try it anyway since it is not one of those whose answers are included in the back of the book. I know that when I was in school, my procedure was to avoid the keyless problems like the plague. I was, needless to say, a clueless student who came away from the course not much better off than before I came into it. It took me a long long time to learn how to learn.

Teachers can use problems like this to see which students are serious...they are the ones who will come up after class and ask how to proceed with these mavericks.