How to tell how many solutions c^2 - 18 = 9 in mental math?

a quadratic always has two solutions.

The answer key says 3+-square root 3

I think you mean

±3√3

The two solutions are √27 and -√27

But, √27 = √(9*3) = √9√3 = 3√3

Oh, okay. Thanks!

To determine the number of solutions of the equation c^2 - 18 = 9 mentally, we need to rearrange the equation and use mental calculations to solve it. Here are the steps:

1. Start by adding 18 to both sides of the equation: c^2 = 27.

2. To find the solutions, take the square root of both sides of the equation: √(c^2) = √27.

3. Simplifying the equation, we have |c| = √(9*3).

4. Since the square root of 9 is 3, we can rewrite the equation as |c| = 3√3.

5. Now, consider the equation |c| = 3√3. The absolute value of c will always be positive or zero, so we can drop the absolute value sign.

6. The equation simplifies to c = 3√3.

Thus, we have determined that the equation c^2 - 18 = 9 has only one solution, which is c = 3√3.