To solve the quadratic equation \( x^2 - 5x + 10 = 0 \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this equation:
- \( a = 1 \)
- \( b = -5 \)
- \( c = 10 \)
Now, plug these values into the formula:
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Calculate the discriminant \( b^2 - 4ac \): \[ (-5)^2 - 4 \times 1 \times 10 = 25 - 40 = -15 \]
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Since the discriminant is negative, we will get complex solutions.
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Now use the quadratic formula: \[ x = \frac{-(-5) \pm \sqrt{-15}}{2 \times 1} = \frac{5 \pm \sqrt{-15}}{2} \]
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We can express \( \sqrt{-15} \) as \( i\sqrt{15} \), where \( i \) is the imaginary unit.
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Therefore, the solutions can be simplified to: \[ x = \frac{5 \pm i\sqrt{15}}{2} \]
Thus, the solutions to the equation \( x^2 - 5x + 10 = 0 \) are:
\[ x = \frac{5}{2} \pm \frac{i\sqrt{15}}{2} \]
This can be expressed as:
\[ x = \frac{5}{2} \pm \frac{\sqrt{15}}{2} i \]