# how to solve without solving

• Dec 6th 2006, 11:49 AM
shenton
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.
• Dec 6th 2006, 01:40 PM
topsquark
Quote:

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
• Dec 6th 2006, 03:28 PM
Soroban
Hello, shenton!

Actually, you had it, Dan . . .

Quote:

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: $x^2 - 2x - 9$

Let $a$ and $b$ be the roots of the equation.
Then: . $a + b \,=\,4,\;ab\,=\,-6$

Let the roots of the new equation be: . $\begin{array}{cc}A\:=\:a-1 \\ B\:=\:b-1\end{array}$

Then we have:
. . $A + B \;=\; (a-1)+(b-1) \;= \; (a + b) - 2 \;=\; 4 - 2\quad\Rightarrow\quad \boxed{A+B\:=\:2}$
$AB \;= \; (a-1)(b-1) \;=\; ab - (a+b) + 1 \;=\; -6 - 4 + 1\quad\Rightarrow\quad \boxed{AB\:=\:-9}$

Therefore, the equation is: . $\boxed{x^2 -2x - 9\:=\:0}$