Why does the equation 3x^2 - 30 = 3 have two solutions but the equation 3x^2 + 30 = 3 has no solution?

Question

Try some different values for x

Answer 1

By solving the equation 3x^2 - 30 = 3:

3x^2 - 30 = 3
3x^2 = 33
x^2 = 11
x = ±√11

Thus, the equation has two solutions.

By solving the equation 3x^2 + 30 = 3:

3x^2 + 30 = 3
3x^2 = -27

Since the square of any real number is non-negative, there is no real number solution for this equation. This is why the equation has no solution.

To confirm, let's try some different values for x:

For the first equation:
If x = √11, then 3(√11)^2 - 30 = 3, which is true.
If x = -√11, then 3(-√11)^2 - 30 = 3, which is true.

Both solutions fulfill the first equation.

For the second equation:
If x = √11, then 3(√11)^2 + 30 = 3 + 30 = 33, which is not true.
If x = -√11, then 3(-√11)^2 + 30 = 3 + 30 = 33, which is not true.

Neither solution fulfills the second equation.

Therefore, the solutions for the equation 3x^2 - 30 = 3 are x = ±√11, while the equation 3x^2 + 30 = 3 has no real number solutions.