The expression ( 9a^2 + 24a + 16 ) is a quadratic trinomial. To understand it better, you might want to factor it or find its roots.
To factor it, we can look for two numbers that multiply to ( 9 \times 16 = 144 ) and add to ( 24 ). The numbers ( 12 ) and ( 12 ) meet these criteria.
Now, we can rewrite the expression:
[ 9a^2 + 12a + 12a + 16 ]
Next, we can group the terms:
[ (9a^2 + 12a) + (12a + 16) ]
Factoring by grouping, we have:
[ 3a(3a + 4) + 4(3a + 4) ]
Now, we can factor out the common binomial factor ( (3a + 4) ):
[ (3a + 4)(3a + 4) = (3a + 4)^2 ]
So, the complete factorization of ( 9a^2 + 24a + 16 ) is:
[ (3a + 4)^2 ]
If you're looking for the roots of the equation ( 9a^2 + 24a + 16 = 0 ), you can set the factored expression to zero:
[ (3a + 4)^2 = 0 ]
This gives:
[ 3a + 4 = 0 ]
[ 3a = -4 ]
[ a = -\frac{4}{3} ]
Thus, the roots of the equation are ( a = -\frac{4}{3} ) (with multiplicity 2).
In summary:
- The expression ( 9a^2 + 24a + 16 ) factors to ( (3a + 4)^2 ).
- The root is ( a = -\frac{4}{3} ).
