3.

Question

Use the quadratic formula to solve the equation. If necessary, round to the nearest hundredth.

a2 − 2a − 224 = 0

8, –28

–8, 28

16, –14

–16, 14

Answer 1

To solve the quadratic equation $a^2-2a-224=0$ using the quadratic formula, we will follow these steps:

The quadratic formula is given by:

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

For the equation $a^2-2a-224=0$, we identify the coefficients as follows:

- $a=1$
- $b=-2$
- $c=-224$

Now we plug these values into the quadratic formula:

$$a=\frac{-(-2)\pm \sqrt{(-2{)}^2-4\cdot 1\cdot (-224)}}{2\cdot 1}$$

Simplifying inside the square root:

$$a=\frac{2\pm \sqrt{4+896}}{2}$$

$$a=\frac{2\pm \sqrt{900}}{2}$$

We know that $\sqrt{900}=30$, so:

$$a=\frac{2\pm 30}{2}$$

This gives us two solutions:

1. When we take the positive root:

$$a=\frac{2+30}{2}=\frac{32}{2}=16$$

2. When we take the negative root:

$$a=\frac{2-30}{2}=\frac{-28}{2}=-14$$

So the solutions are $a=16$ and $a=-14$.

Thus, the correct answer is:
$$\boxed{{\displaystyle 16,-14}}$$