Let y be equal to 2a-3 and rewrite the expression in terms of y. Factor the new expression, and then substitute 2a-3 for y.
2(a+1)(6a-7)
Practice makes perfect
We are asked to factor the given expression.
3(2a-3)^2 + 17(2a-3)+10
Notice that the expression (2a-3) is common for the first two terms. We can make a variable change. Let's call y= 2a-3. Then, we can rewrite the expression in terms of y.
3( 2a-3)^2 + 17( 2a-3) + 10= 3 y^2 + 17 y + 10
We obtained a quadratic expression in the form of dy^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 d c and have a sum of b.
d c = 3(10)=30 b = 17
Since d c> 0, the factors have the same sign. Moreover, because b> 0, the factors are both positive.
Factors of 30
Sum
1, 30
1+30=31 *
2,15
2+15=17 âś“
3, 10
3+10=13 *
5, 6
5+6=11 *
Using the factors 2 and 15, we can rewrite the expression as follows.
3y^2 + 17y + 10 = 3y^2 + 15y + 2y_(17y) + 10
From the latter expression, we factor out 3y from the first two terms and 2 from the last two.
3y^2+15y +2y + 10 = 3y(y+5) + 2(y+5)
Now, we can factor out (y+5).
3y(y+5) + 2(y+5) = (y+5)(3y+2)
Finally, we can revert back the variable change we did at the beginning. To do so, we substitute y= 2a-3 and simplify the expression.
( y+5)(3 y+2) &= ( 2a-3+5)(3( 2a-3)+2)
&= (2a+2)(6a-7)
Notice that we can factor our 2 from the first factor in the expression above. After that, we will have factored the expression completely.
(2a+2)(6a-7) = 2(a+1)(6a-7)
