The errors include:
- Incorrect square root calculation
- Incorrect sign for square root term
- Incorrect value for discriminant
- Ignoring the imaginary unit when taking the square root of a negative number
There are several errors in the provided solution to the equation 3x^2 - 4x = 20:
1. Incorrect square root calculation:
The calculation of the square root in the equation is incorrect. The negative sign should be inside the square root, not outside. It should read:
x = (-4 ± sqrt(-4^2 - 4ac)) / (2a)
2. Incorrect sign for square root term:
In the subsequent step, the wrong sign is applied to the square root term. It should be:
x = (-4 ± √(-4)^2 - 4(3)(20)) / (2a)
3. Incorrect value for discriminant:
Following the equation, the discriminant (-4)^2 - 4(3)(20) should be evaluated as -256 instead of -16 - 240.
4. Incorrect square root of a negative number:
The square root of a negative number is imaginary and cannot be represented by a real number. Therefore, the equation has no real solutions.
5. Incorrect conclusion:
The statement "No Solutions" is correct because the equation has no real solutions due to the negative discriminant.
In summary, the errors include:
Incorrect square root calculation
Incorrect sign for square root term
Incorrect value for discriminant
Ignoring the imaginary unit when taking the square root of a negative number
These errors collectively lead to the incorrect conclusion that the equation has no real solutions, which is indeed correct.