Determine the number of real solutions there are for `9x^2-12x+18=0`

Options

A. no real solution
B. 1 real solution
C. 2 real solution
D. 3 real solution

Answer

To determine the number of real solutions for the quadratic equation \(9x^2 - 12x + 18 = 0\), we can use the discriminant method. The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula:

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

### Step 1: Identify the coefficients
From the equation \(9x^2 - 12x + 18 = 0\), we identify:
- \(a = 9\)
- \(b = -12\)
- \(c = 18\)

### Step 2: Calculate the discriminant
Now we will calculate the discriminant \(D\):

\[ D = (-12)^2 - 4 \cdot 9 \cdot 18 \]

Calculating \(b^2\):

\[ (-12)^2 = 144 \]

Calculating \(4ac\):

\[ 4 \cdot 9 \cdot 18 = 648 \]

Now substituting these values into the discriminant formula:

\[ D = 144 - 648 \]

Calculating \(D\):

\[ D = 144 - 648 = -504 \]

### Step 3: Analyze the discriminant
The value of the discriminant \(D\) is \(-504\). Since the discriminant is negative, it indicates that there are no real solutions to the quadratic equation.

### Conclusion
Thus, the number of real solutions for the equation \(9x^2 - 12x + 18 = 0\) is **0**.

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