To solve the quadratic equation \(x^2 - 6x + 25 = 0\) using the quadratic formula, we can follow these steps:
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Identify coefficients: The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients from the equation \(ax^2 + bx + c = 0\).
For the equation \(x^2 - 6x + 25 = 0\):
- \(a = 1\)
- \(b = -6\)
- \(c = 25\)
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Calculate the discriminant: We need to calculate \(b^2 - 4ac\): \[ b^2 - 4ac = (-6)^2 - 4 \cdot 1 \cdot 25 = 36 - 100 = -64 \]
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Apply the quadratic formula: Since the discriminant is negative, this means the solutions will be complex: \[ x = \frac{-(-6) \pm \sqrt{-64}}{2 \cdot 1} = \frac{6 \pm \sqrt{-64}}{2} \] \[ \sqrt{-64} = 8i \] So the equation becomes: \[ x = \frac{6 \pm 8i}{2} = 3 \pm 4i \]
Therefore, the solutions to the equation \(x^2 - 6x + 25 = 0\) are \(3 + 4i\) and \(3 - 4i\).
The correct choice from the options provided is: C \(3 \pm 4i\).
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