Which of the following equations does NOT have any solutions that are real numbers ?

Options

A. `x ^(2)+8x-12=0`
B. `x ^(2) -8x +12 =0`
C. `x ^(2) -9x +21 =0`
D. `x ^(2) + 100x -1 =0`

Explanation

To determine which of the following quadratic equations does NOT have any solutions that are real numbers, we will use the discriminant method. The discriminant \( D \) for a quadratic equation of the form \( ax^2 + bx + c = 0 \) is given by:

\[ D = b^2 - 4ac \]

If \( D < 0 \), the quadratic equation has no real solutions. We will evaluate the discriminant for each of the given options.

### Step 1: Evaluate the first option
Assume the first option is \( x^2 + 8x - 12 = 0 \).

- Here, \( a = 1 \), \( b = 8 \), and \( c = -12 \).
- Calculate the discriminant:

\[ D = b^2 - 4ac = 8^2 - 4 \cdot 1 \cdot (-12) = 64 + 48 = 112 \]

Since \( D = 112 > 0 \), this equation has real solutions.

### Step 2: Evaluate the second option
Assume the second option is \( x^2 - 8x + 12 = 0 \).

- Here, \( a = 1 \), \( b = -8 \), and \( c = 12 \).
- Calculate the discriminant:

\[ D = b^2 - 4ac = (-8)^2 - 4 \cdot 1 \cdot 12 = 64 - 48 = 16 \]

Since \( D = 16 > 0 \), this equation also has real solutions.

### Step 3: Evaluate the third option
Assume the third option is \( x^2 - 9x + 21 = 0 \).

- Here, \( a = 1 \), \( b = -9 \), and \( c = 21 \).
- Calculate the discriminant:

\[ D = b^2 - 4ac = (-9)^2 - 4 \cdot 1 \cdot 21 = 81 - 84 = -3 \]

Since \( D = -3 < 0 \), this equation does NOT have real solutions.

### Conclusion
The quadratic equation that does NOT have any solutions that are real numbers is the third option.

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