Let y be equal to x-2 and rewrite the expression in terms of y. Factor the new expression, and then substitute x-2 for y.
(x-9)(x-10)
Practice makes perfect
We are asked to factor the given expression.
(x-2)^2 - 15(x-2)+56
Notice that the expression (x-2) is common for the first two terms. We can make a variable change. Let's call y= x-2. Then, we can rewrite the expression in terms of y.
( x-2)^2 - 15( x-2) + 56 = y^2 - 15 y + 56
We obtained a quadratic expression in the form of y^2 + by + c on the right-hand side. Now, we will factor it. Since it has no common factors, we will try to rewrite b as the sum of two terms with coefficients that are factors of c and have a sum of b.
c = 56 b = - 15
Since c> 0, the factors have the same sign. Moreover, because b< 0, the factors are both negative.
Factors of 56
Sum of Factors
- 1, - 56
- 1+(- 56)=- 57 *
- 2, - 28
- 2+(- 28)=- 30 *
- 4,- 14
- 4+(- 14)=- 18 *
- 7,- 8
- 7+(- 8)=- 15 âś“
Using the factors - 7 and - 8, we can rewrite the expression as follows.
y^2 - 15y + 56 = y^2 - 7y - 8y_(- 15y) + 56
From the latter expression, we factor out y from the first two terms and - 8 from the last two.
y^2- 7y - 8y + 56 = y(y-7) - 8(y-7)
Now, we can factor out (y-7).
y(y-7) - 8(y-7) = (y-7)(y-8)
Finally, we can revert back the variable change we did at the beginning. To do so, we substitute y= x-2 and simplify the expression.
( y-7)( y-8) &= ( x-2-7)( x-2-8)
&= (x-9)(x-10)
