Homework Help: Expressing c in terms of b

1. Nov 1, 2009

um0123

1. The problem statement, all variables and given/known data

The roots of $$x^2 + bx + c = 0$$ are $$r_1$$ and $$r_2$$,
where $$|r_1 - r_2| = 1$$.

Express c in terms of b.

2. Relevant equations

$$x^2 + bx + c = 0$$

$$|r_1 - r_2| = 1$$

3. The attempt at a solution
Im not even sure where to start with this problem, and i dont see how the roots are relevant. I just dont know what the question is asking.

2. Nov 1, 2009

danago

Can you write r1 and r2 in terms of b and c? i.e. actually find the roots.

The condition then says that the difference between these roots must be 1. Using the roots that you found above, set their difference equal to one. When does this equation hold true? How must b and c be related to satisfy this equation?

3. Nov 1, 2009

Staff: Mentor

The roots are very relevant. If r1 and r2 are roots of your quadratic equation, it can be written as (x - r1)(x - r2) = 0.

Multiply the product above and equate the coefficient of x with b, and the constant term above with c.

The other condition, |r1 - r2| = 1, says that either r1 = r2 + 1 or that r2 = r1 + 1.

4. Nov 1, 2009

um0123

(x - r1)(x - r2) =

$$x^2 -r_1x - r_2x -r_1r_2$$

so how do i combine the middle terms to get a single coefficient for b.

5. Nov 1, 2009

danago

Take out a factor of x:

$$(x - r_1)(x - r_2) = x^2 -x(r_1+r_2) +r_1r_2$$

Also, be careful with your signs. The constant term should be +r1r2

6. Nov 1, 2009

um0123

okay, im seeing it a little more now, but i still dont understand how to relate b and c. I guess you can say that c is the product and b is the sum (of the values r1 and r2), but thats not relating the two to each other. And i still dont understand why its important that the absolute value is equal to 1.

7. Nov 1, 2009

danago

It is important because that is the condition that the question requires. Think about when you normally find the solutions to a quadratic equation; you will (normally) get two unique solutions? What the question is saying is that when you solve this particular one, the difference between these two roots is 1.

If you use the quadratic formula to solve the given equation, what two roots do you get?

8. Nov 1, 2009

um0123

So i have to use the quadratic equation on $$x^2 -x(r_1+r_2) +r_1r_2$$

x = -(r_1+r_2) +- sqrt{(r_1+r_2)^2 - 4(r_1r_2)}/2

that hardly seems solvable, i know im doing something wrong.

9. Nov 1, 2009

danago

Apply it to the original equation to find x in terms of b and c.

10. Nov 1, 2009

um0123

sorry, i dont understand what you mean by apply it to the original equation.

11. Nov 1, 2009

danago

$$x^2 + bx + c = 0$$

Solving this gives:
$$x=\frac{-b+\sqrt{b^2-4c}}{2}$$

and

$$x=\frac{-b-\sqrt{b^2-4c}}{2}$$

The condition says that the difference between these is 1.

12. Nov 1, 2009

um0123

i understand that part, but i dont understand how i solve it. Can i use a systems of equations?

13. Nov 1, 2009

danago

Well the $$|r_1 - r_2| = 1$$ part says that the difference between these two solutions must be 1.

How about you actually find the difference between the two solutions above and then determine what the relationship between b and c must be for this difference to equal 1.

14. Nov 1, 2009

um0123

so

$$\frac{-b+\sqrt{b^2-4c}}{2} - \frac{-b-\sqrt{b^2-4c}}{2}=1$$

multiply each side by 2

$$-b+\sqrt{b^2-4c} - {-b-\sqrt{b^2-4c}=2$$

simplify

$$-b+\sqrt{b^2-4c} + b +\sqrt{b^2-4c}=2$$

-b and b cancel

$$\sqrt{b^2-4c}+\sqrt{b^2-4c}=2$$

square both sides

$$b^2-4c+b^2-4c=4$$

simplify

$$2b^2-8c=4$$

divide everything by 2

$$b^2-4c=2$$

add 4c to both sides and subtract 2 from both sides

$$b^2-2=4c$$

divide by 4 on both sides

$$\frac{b^2-2}{4}=c$$

is that all i can do? did i do it right?

or am i supposed to solve for b?

$$b^2-4c=2$$

$$b^2=2+4c$$

square root both sides

$$b=\sqrt{2+4c}$$

Last edited: Nov 1, 2009
15. Nov 1, 2009

um0123

Sorry for the bump, but i am worried i did this wrong, can someone tell me if i did this correctly?

16. Nov 1, 2009

danago

Have another look at that step where you squared both sides.

In general, it is NOT true that $$(a+b)^2=a^2+b^2$$

17. Nov 2, 2009

um0123

in that case i dont know what to do....

18. Nov 2, 2009

danago

You had the right idea, just a small error in the algebra.

$$\sqrt{b^2-4c}+\sqrt{b^2-4c}= 2\sqrt{b^2-4c}=2$$

How about divide both sides by 2?

$$\sqrt{b^2-4c}=1$$

Can you see what to do from there? For that equation to hold, how must b and c be related?

19. Nov 2, 2009

um0123

omg, thank you so much!!!!!!

20. Nov 2, 2009

danago

So you see it now?

Glad to be of assistance :)