Proving Equal Roots in ar^2+br+c=0 with L[e^(rt)]

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In summary, for an equation with equal roots r1, the first and second derivatives of e^(rt) can be represented as a combination of a, b, and c. Furthermore, if r1 is a double root, the equation can be rewritten as a{(r-r1)^2}e^(rt) with the given coefficients. Advice was given to utilize the first and second derivatives of e^(rt) to solve the equation on the left hand side.
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asdf1
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If ar^2+br+c=0 has equal roots r1, show that
L[e^(rt)]=a(e^(rt))``+b(e^(rt))`+ ce^(rt)=a{(r-r1)^2}e^(rt)

could someone offer some advice?
 
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asdf1 said:
If ar^2+br+c=0 has equal roots r1, show that
L[e^(rt)]=a(e^(rt))``+b(e^(rt))`+ ce^(rt)=a{(r-r1)^2}e^(rt)
could someone offer some advice?
The only advice I can give is that you go ahead and do what is shown on the left hand side!
Surely, you know what the first and second derivatives of ert are!
And, of course, If r1 is a double root of ar2+ br+ c= 0, then ar2+ br+ c= a(r- r1[/sup])(r-r2).
 
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ok ~ thanks!
 

1. How can I prove that the roots of the equation ar^2+br+c=0 are equal?

In order to prove that the roots of this quadratic equation are equal, we can use the discriminant formula: b^2-4ac=0. If the discriminant is equal to zero, then the roots will be equal.

2. Can I use the quadratic formula to prove equal roots?

Yes, the quadratic formula (x = (-b ± √(b^2-4ac)) / 2a) can also be used to prove that the roots are equal. If the value inside the square root is equal to zero, then the roots will be equal.

3. Are there any other methods to prove equal roots in this equation?

Yes, another method is to factor the equation and see if the resulting factors have a common root. If they do, then the original equation has equal roots.

4. Why is it important to prove equal roots in this equation?

Proving equal roots in this equation is important because it helps us determine the behavior and properties of the graph of the equation. If the roots are equal, then the graph will have a double root or a point of inflection at that value.

5. Can we apply this concept to other types of equations?

Yes, the concept of proving equal roots can be applied to other types of equations as well, such as cubic or quartic equations. We can use similar methods, such as the discriminant or factoring, to determine if the roots are equal in these equations.

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