Solve the equation 3a^2 + 24a = -45.

Step-by-step solution

1. 1

   Bring the equation to standard form. Add $45$ to both sides:

   $3a^2+24a+45=0$

2. 2

   Divide through by the common factor. All three coefficients are divisible by $3$, and dividing an equation by a nonzero constant does not change its roots:

   $a^2+8a+15=0$

   This is worth doing before factoring — a leading coefficient of $1$ makes the search for factors much easier.

3. 3

   Factor the monic quadratic. Two numbers with product $15$ and sum $8$ are $3$ and $5$:

   $(a+3)(a+5)=0$

4. 4

   Use the zero-product property.

   $a=-3\qquad\text{or}\qquad a=-5$

5. 5

   Check in the original equation, not the reduced one. At $a=-3$: $3(9)+24(-3)=27-72=-45$ $\checkmark$. At $a=-5$: $3(25)-120=75-120=-45$ $\checkmark$. Vieta also agrees: the roots sum to $-8$ and multiply to $15$.