# Quadratic Simultaneous Eqns: ""J and T share money. If I square J's money and add..."

#### Jim77

##### New member
Quadratic Simultaneous Eqns: ""J and T share money. If I square J's money and add..."

Hi all

Can someone give me a hint on how to solve the below please?

"Jim and Tim are sharing money. If I square Jim's money and add on Tim's, I get £10,050. If I square Tim's money and add on Jim's, I get £2,600. How much money does each have?"

Regards

Jim

#### stapel

##### Super Moderator
Staff member
Can someone give me a hint on how to solve the below please?

"Jim and Tim are sharing money. If I square Jim's money and add on Tim's, I get £10,050. If I square Tim's money and add on Jim's, I get £2,600. How much money does each have?"
A good place to start would probably be with picking variables to stand for the unknowns. What variables did you pick, and how did you define them?

Thank you! j^2 + t = 10,050
t^2 + j = 2,600

#### stapel

##### Super Moderator
Staff member
j^2 + t = 10,050
t^2 + j = 2,600
Okay; you've jumped ahead to equations. These equations contain the variables "j" and "t". From the way you've used the variables, I believe you defined them as follows:

. . .variables:
. . . . .Tim's money: t
. . . . .Jim's money: j

If this is correct, then your equations look to be accurate reflections of the relationships given in the exercise.

So now you have two equations in two unknowns. What method(s) have you been taught for solving systems of non-linear equations? For instance, are you familiar with "substitution"? If you are, how might that tool be applied to this system? Where does this lead? #### Jim77

##### New member
Rearrange the second equation to give:

j = 2600 - T^2

Is this correct?

anyone?

#### JeffM

##### Elite Member
Rearrange the second equation to give:

j = 2600 - T^2

Is this correct?
Yes, this is correct.

This allows you to restate the other equation how?

By the way, most of the volunteers here, who are paid not a penny, live in North America, which is at least 5 hours behind time in the UK and may be 10 hours behind. Unpaid volunteers are not necessarily going to leap from bed on a Sunday morning to work on math problems.

#### Jim77

##### New member
By the way, most of the volunteers here, who are paid not a penny, live in North America, which is at least 5 hours behind time in the UK and may be 10 hours behind. Unpaid volunteers are not necessarily going to leap from bed on a Sunday morning to work on math problems.
Apparently so...

so (2600 - t^2)^2 + t = 10,050

Is this correct?

This is where I have no idea on how to solve for t...

#### Jonathan

##### New member
j^2 + t = 10,050
t^2 + j = 2,600
I didn't get far using substitution. Try subtracting the second equation from the first.

#### Jim77

##### New member
I didn't get far using substitution. Try subtracting the second equation from the first.
Sorry could you expand on that a little bit please?

#### Subhotosh Khan

##### Super Moderator
Staff member
Apparently so...

so (2600 - t^2)^2 + t = 10,050

Is this correct?

This is where I have no idea on how to solve for t...
(2600 - t^2)^2 + t = 10,050

Expand

t^4 - 5200* t^2 + t + 6749950 = 0

In a previous post you were asked:

What method(s) have you been taught for solving systems of non-linear equations?
Please respond to that question!!

#### Jonathan

##### New member
Sorry could you expand on that a little bit please?
The admins of this site are not happy if a full solution is given early, so I can only give hints.
In his reply Subhotosh Khan is assuming that this question is an exercise in using substitution and wants you to explain what you have learnt so far.
I believe that that there is a neater solution.

(1) j^2 + t = 10,050
(2) t^2 + j = 2,600
Make a new equation by subtracting terms of (1) from those of (2)
j^2 + t - t^2 - j = 10,050 - 2,600
see whether you can anything of this

#### Jonathan

##### New member
Yo Jonathan!
That'll simplify to: 2j = 1 +- SQRT(4t^2 - 4t + 29801)
So (I think!) you'll still end up with a 4th degree...
Or do I need another coffee j^2 + t = 10,050
Btw, the value of j is "evident" from just "looking" at that equation.
Closest square < 10050
I'm still not sure about how much one can give away immediately. I think that unless we know that the question in the OP was homework to practice substitution there's little point pursuing that path. Especially as it leads to to an awkward quartic.
In j^2 - t^2 + t - y = 10050 - 2600, I grouped the expressions on the left, saw "difference of two squares" so then could factorise with just two factors.
Saw which pair of values matched the factors of 10050 - 2600.
Must admit I assumed a whole number of dollars.

#### JeffM

##### Elite Member
I'm still not sure about how much one can give away immediately. I think that unless we know that the question in the OP was homework to practice substitution there's little point pursuing that path. Especially as it leads to to an awkward quartic.
In j^2 - t^2 + t - y = 10050 - 2600, I grouped the expressions on the left, saw "difference of two squares" so then could factorise with just two factors.
Saw which pair of values matched the factors of 10050 - 2600.
Must admit I assumed a whole number of dollars.
$$\displaystyle (j^t + t^2) + t - j = 10050 - 2600 \implies (j - t)(j + t) - (j - t) = (j - t)(j + t - 1) = 7450.$$

I agree that your factoring is valid and can be used to solve the problem by extensive trial and error.

But if you are going to assume whole numbers you can use the integer root theorem and get an answer very efficiently from the quartic.

$$\displaystyle t^4 - 5200t^2 + t + 6,749,950 = 0.$$ I have no idea why this does not render properly.

Factor the constant term with the absolutely obvious factor of 10

$$\displaystyle 6,749,950 = 10 *674,995.$$

Factor again by the obvious factor of 5

$$\displaystyle 6,749,950 = 10 * 5 * 134999.$$

It is obvious virtually by inspection that 2, 5, 10, and 25 are too small. 25 squared is 625 which would make j almost 2000, and that squared is way bigger than 10050. So try t = 50.

$$\displaystyle 50^4 - 5200 * 50^2 + 50 = 6,250,000 - 50 * 2500 + 50 =$$

$$\displaystyle 6,250,050 - 13,000,000 = -\ 6,749,950.$$

50 works.

I'd be amazed if this problem was not found in a section that mentions either the rational or integer root theorems or both.

Last edited:

#### JeffM

##### Elite Member
j^2 + t = 10,050

j = sqrt(10000) = 100
t = 10050 - 10000 = 50

Ahem Mon ami, I do not disagree that applying the answer suggested by the rational root theorem into the original equations is even more efficient. That, however, does not teach a lot about the potential of the theorem.

#### JeffM

##### Elite Member
Or use Newton's method - converges in about 3 steps.
I suspect someone who does not know the rational root theorem and its corollary of the integer root theorem is not quite up to Newton's method for finding the zeroes of a polynomial. But it's definitely a method. And of course there is always a graphing calculator.

#### Subhotosh Khan

##### Super Moderator
Staff member
It could be a problem to teach follies of Newton's method.

If you start at 51 - it converges to 51.96143

If you start at 50.9 - it converges to 50

#### Jonathan

##### New member
$$\displaystyle (j^t + t^2) + t - j = 10050 - 2600 \implies (j - t)(j + t) - (j - t) = (j - t)(j + t - 1) = 7450.$$

I agree that your factoring is valid and can be used to solve the problem by extensive trial and error.

But if you are going to assume whole numbers you can use the integer root theorem and get an answer very efficiently from the quartic.

$$\displaystyle t^4 - 5200t^2 + t + 6,749,950 = 0.$$ I have no idea why this does not render properly.

Factor the constant term with the absolutely obvious factor of 10

$$\displaystyle 6,749,950 = 10 *674,995.$$

Factor again by the obvious factor of 5

$$\displaystyle 6,749,950 = 10 * 5 * 134999.$$

It is obvious virtually by inspection that 2, 5, 10, and 25 are too small. 25 squared is 625 which would make j almost 2000, and that squared is way bigger than 10050. So try t = 50.

$$\displaystyle 50^4 - 5200 * 50^2 + 50 = 6,250,000 - 50 * 2500 + 50 =$$

$$\displaystyle 6,250,050 - 13,000,000 = -\ 6,749,950.$$

50 works.

I'd be amazed if this problem was not found in a section that mentions either the rational or integer root theorems or both.
I reduced the equation to
(j−t)(j+t−1) = 2 x 5 x 5 x 149
and spotted factors 50 x 149 "by inspection", it did not require "extensive trial and error".
At least as quick as all the steps above.

#### JeffM

##### Elite Member
I reduced the equation to
(j−t)(j+t−1) = 2 x 5 x 5 x 149
and spotted factors 50 x 149 "by inspection", it did not require "extensive trial and error".
At least as quick as all the steps above.
Fair enough. But you had said you did not get very far with substitution. Numerous methods work after substitution.

#### Jim77

##### New member
I'm still not sure about how much one can give away immediately. I think that unless we know that the question in the OP was homework to practice substitution there's little point pursuing that path. Especially as it leads to to an awkward quartic.
In j^2 - t^2 + t - y = 10050 - 2600, I grouped the expressions on the left, saw "difference of two squares" so then could factorise with just two factors.
Saw which pair of values matched the factors of 10050 - 2600.
Must admit I assumed a whole number of dollars.
Hi Jonathan - so you think this is the most straightforward approach?

I know that the difference of two squares can be expressed as (j+t)(t-j) or (j-t)(t+j) ? Is this correct?