How to Express c in Terms of b in Quadratic Equations?

[um0123]

Homework Statement

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.

Homework Equations

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

$$|r_1 - r_2| = 1$$

The Attempt at a Solution

Im not even sure where to start with this problem, and i don't see how the roots are relevant. I just don't know what the question is asking.

 

[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?

 

[Mark44]

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.

 

[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.

 

[danago]

(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.

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

 

[um0123]

okay, I am seeing it a little more now, but i still don't 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 that's not relating the two to each other. And i still don't understand why its important that the absolute value is equal to 1.

 

[danago]

okay, I am seeing it a little more now, but i still don't 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 that's not relating the two to each other. And i still don't understand why its important that the absolute value is equal to 1.

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?

 

[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 I am doing something wrong.

 

[danago]

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

 

[um0123]

sorry, i don't understand what you mean by apply it to the original equation.

 

[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.

 

[um0123]

i understand that part, but i don't understand how i solve it. Can i use a systems of equations?

 

[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.

 

[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?

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

add 4c to both sides

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

square root both sides

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

 

[um0123]

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

 

[danago]

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

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

 

[um0123]

in that case i don't know what to do...

 

[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?

 

[um0123]

omg, thank you so much!

 

[danago]

So you see it now?

Glad to be of assistance :)