To find the discriminant of the quadratic equation \(3r^2 - 6r + 6 = 0\), we can use the formula for the discriminant, which is given by:
\[ D = b^2 - 4ac \]
In the given equation \(3r^2 - 6r + 6 = 0\), we identify the coefficients:
- \(a = 3\)
- \(b = -6\)
- \(c = 6\)
Now, we can substitute these values into the discriminant formula:
\[ D = (-6)^2 - 4(3)(6) \]
Calculating this step by step:
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Calculate \(b^2\): \[ (-6)^2 = 36 \]
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Calculate \(4ac\): \[ 4(3)(6) = 72 \]
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Now substitute back into the formula: \[ D = 36 - 72 = -36 \]
The discriminant \(D = -36\).
Since the discriminant is negative, this means that the quadratic equation has no real solutions. Instead, it has two complex (or imaginary) solutions.
Summary:
- Discriminant \(D = -36\)
- The equation has 0 real solutions (2 complex solutions).