2. Why does the equation 3x2 - 30 = 3 have two solutions but the equation 3x2 + 30 = 3 has no solution?
Try some different values for x.
To determine the number of solutions for a quadratic equation, we can consider the discriminant. The discriminant is the term inside the square root of the quadratic formula and is given by b^2 - 4ac.
For the equation 3x^2 - 30 = 3, the coefficients are a = 3, b = 0, and c = -27.
Plugging these values into the discriminant formula, we get discriminant = 0^2 - 4(3)(-27) = 0 - (-324) = 324.
Since the discriminant is positive (324 > 0), the equation has two real and distinct solutions.
Now let's consider the equation 3x^2 + 30 = 3. The coefficients are a = 3, b = 0, and c = 27.
Plugging these values into the discriminant formula, we get discriminant = 0^2 - 4(3)(27) = 0 - (324) = -324.
Since the discriminant is negative (-324 < 0), the equation has no real solutions.
To further understand this, let's try some different values for x:
If we substitute x = 1 into 3x^2 - 30 = 3, we get 3(1)^2 - 30 = 3, which simplifies to 3 - 30 = 3, or -27 = 3. This is not true, so x = 1 is not a solution.
If we substitute x = 1 into 3x^2 + 30 = 3, we get 3(1)^2 + 30 = 3, which simplifies to 3 + 30 = 3, or 33 = 3. This is also not true, so x = 1 is not a solution.
This demonstrates that the equation 3x^2 - 30 = 3 has no solutions, as the x-values we tested did not satisfy the equation. On the other hand, the equation 3x^2 + 30 = 3 also has no solutions, as the x-values we tested did not satisfy the equation either.