9а²+24а+16

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 \( a = -\frac{4}{3} \).