To solve the quadratic equation \( x^2 + 3x - 6 = 0 \) using the quadratic formula, we will use the formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this equation \( a = 1 \), \( b = 3 \), and \( c = -6 \).
First, we calculate the discriminant (\( b^2 - 4ac \)):
\[ b^2 = 3^2 = 9 \] \[ 4ac = 4 \cdot 1 \cdot (-6) = -24 \] \[ b^2 - 4ac = 9 - (-24) = 9 + 24 = 33 \]
Now, we substitute the values into the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{33}}{2 \cdot 1} \] \[ x = \frac{-3 \pm \sqrt{33}}{2} \]
This gives us the two solutions:
\[ x = \frac{-3 + \sqrt{33}}{2} \quad \text{and} \quad x = \frac{-3 - \sqrt{33}}{2} \]
So the solutions to the equation \( x^2 + 3x - 6 = 0 \) are:
\[ x = \frac{-3 + \sqrt{33}}{2} \quad \text{or} \quad x = \frac{-3 - \sqrt{33}}{2} \]
If you want numerical approximations:
\[ x \approx 0.372 \quad \text{and} \quad x \approx -6.372 \]
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