Question

If a^2+3a+9=0, What is a^3?

Answer 1

Given:

$a^2+3a+9=0$

If we need to find what $a^3$ is, we can solve for a and plug it in. Let's use the quadratic formula to solve for a. The quadratic formula is:

$a=\frac{-b+\sqrt{b^2-4ac}}{2a}$  and  $a=\frac{-b-\sqrt{b^2-4ac}}{2a}$

Let's identify our a, b, and c:

a: 1

b: 3

c: 9

Plug in the values into the quadratic formula:

$a=\frac{-3+\sqrt{(3)^2-4(1)(9)}}{2(1)}$

Simplify everything under the radical:

$a=\frac{-3+\sqrt{-27}}{2}$

Simplify the radical:

$a=\frac{-3+3i\sqrt{3}}{2}$

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Let's solve for the second part of the quadratic formula. Everything will be the same, so we can just replace the + with a -.

$a=\frac{-3+3i\sqrt{3}}{2}$  and  $a=\frac{-3-3i\sqrt{3}}{2}$

These two are your answers. Now, since we know what a is, let's cube this value:

$(a=\frac{-3+3i\sqrt{3}}{2})^3$

Simplify:

$27$

You will get the same answer if you cube both of the quadratic answers. So, your final answer:

$a^3=27$