Let y be equal to x+3 and rewrite the expression in terms of y. Factor the new expression, and then substitute x+3 for y.
(x-3)(x+12)
Practice makes perfect
We are asked to factor the given expression.
(x+3)^2 + 3(x+3)-54
Notice that the expression (x+3) is common for the first two terms. We can make a variable change. Let's call y= x+3. Then, we can rewrite the expression in terms of y.
( x+3)^2 + 3( x+3) - 54 = y^2 + 3 y - 54
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 b.
c = -54 b = 3
Since c< 0, the factors have different signs. Moreover, because b>0, the factor with greatest absolute value is positive.
Factors of -54
Sum of Factors
-1, 54
- 1+54=53 *
-2, 27
- 2+27=25 *
-3,18
- 3+18=15 *
- 6,9
- 6+9=3 âś“
Using the factors - 6 and 9, we can rewrite the expression as follows.
y^2 + 3y - 54 = y^2 - 6y + 9y_(3y) - 54
From the latter expression, we factor out y from the first two terms and 9 from the last two.
y^2- 6y + 9y - 54 = y(y-6) + 9(y-6)
Now, we can factor out (y-6).
y(y-6) + 9(y-6) = (y-6)(y+9)
Finally, we can revert back the variable change we did at the beginning. To do so, we substitute y= x+3 and simplify the expression.
( y-6)( y+9) &= ( x+3-6)( x+3+9)
&= (x-3)(x+12)
