Thread: how to solve without solving

Without solving the given equation, write an equation whose roots could be obtained by subtracting 1 from each root of x^2 - 4x - 6 = 0.

Answer key: x^2 - 2x - 9

How to solve this without solving? Is this asking for Sum of Roots and product of roots?

S = ( ) + ( )
P= ( ) ( )

S = -b/a
P = c/a

I don't think the above 2 sum and product techniques work. What else can I do?

Thanks.

Originally Posted by shenton

Without solving the given equation, write an equation whose roots could be obtained by subtracting 1 from each root of x^2 - 4x - 6 = 0.

Answer key: x^2 - 2x - 9

How to solve this without solving? Is this asking for Sum of Roots and product of roots?

S = ( ) + ( )
P= ( ) ( )

S = -b/a
P = c/a

I don't think the above 2 sum and product techniques work. What else can I do?

Thanks.

Well, the linear term is easy: if we take 1 from each root then the sum of those roots will be decreased by 2:
S' = S - 2
S' = 4 - 2 = 2

The product is a bit tougher and I don't know how to get this result without essentially solving the problem. What I did was this:
The original expression is
(x - a)(x - b)
So we want
(x - (a - 1))(x - (b - 1)) = x^2 - (a + b - 2)x + (ab - (a + b) + 1)

So the last term will be:
P' = P - S + 1
P' = -6 - 4 + 1 = -9
But I really don't know how to prove this result without working it out as above (which seems to be cheating to me.)

-Dan

Hello, shenton!

Actually, you had it, Dan . . .

Without solving the given equation, write an equation whose roots could be obtained
by subtracting 1 from each root of

Answer key:

Let and be the roots of the equation.
Then: .

Let the roots of the new equation be: .

Then we have:
. .


Therefore, the equation is: .