Let y be equal to x+5 and rewrite the expression in terms of y. Factor the new expression, and then substitute x+5 for y.
(2x+9)(3x+14)
Practice makes perfect
We are asked to factor the given expression.
6(x+5)^2 - 5(x+5)+1
Notice that the expression (x+5) is common for the first two terms. We can make a variable change. Let's call y= x+5. Then, we can rewrite the expression in terms of y.
6( x+5)^2 - 5( x+5) + 1= 6 y^2 - 5 y + 1
We obtained a quadratic expression in the form of ay^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 a c and have a sum of b.
a c = 6(1)=6 b = - 5
Since a c> 0, the factors have the same sign. Moreover, because b< 0, the factors are both negative.
Factors of 56
Sum of Factors
- 1, - 6
- 1+(- 6)=- 7 *
- 2,- 3
- 2+(- 3)=- 5 âś“
Using the factors text- 2 and - 3, we can rewrite the expression as follows.
6y^2 - 5y + 1 = 6y^2 - 2y - 3y_(- 5y) + 1
From the latter expression, we factor out 2y from the first two terms and - 1 from the last two.
6y^2- 2y - 3y + 1 = 2y(3y-1) - (3y-1)
Now, we can factor out (3y-1).
2y(3y-1) - (3y-1) = (2y-1)(3y-1)
Finally, we can revert back the variable change we did at the beginning. To do so, we substitute y= x+5 and simplify the expression.
