To solve the quadratic equation \(20x^2 + 20x + 10 = 0\) using the quadratic formula, you can follow these steps:
1. Identify the coefficients: In your equation, \(a = 20\), \(b = 20\), and \(c = 10\).
2. Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
3. Plug in the values:
\(x = \frac{-20 \pm \sqrt{20^2 - 4 \cdot 20 \cdot 10}}{2 \cdot 20}\)
4. Calculate the discriminant:
\(D = b^2 - 4ac = 20^2 - 4 \cdot 20 \cdot 10\)
5. Calculate the solutions:
\(x_1 = \frac{-20 + \sqrt{D}}{2 \cdot 20}\) and \(x_2 = \frac{-20 - \sqrt{D}}{2 \cdot 20}\)
Now, let's compute \(D\) and \(x_1\) and \(x_2\):
\(D = 20^2 - 4 \cdot 20 \cdot 10 = 400 - 800 = -400\)
Since the discriminant (\(D\)) is negative, the quadratic equation has no real solutions. Instead, it has two complex solutions.
So, the solutions for \(x\) are:
\(x_1 = \frac{-20 + \sqrt{-400}}{2 \cdot 20} = \frac{-20 + 20i\sqrt{1}}{2 \cdot 20} = \frac{-1 + i}{2}\)
and
\(x_2 = \frac{-20 - \sqrt{-400}}{2 \cdot 20} = \frac{-20 - 20i\sqrt{1}}{2 \cdot 20} = \frac{-1 - i}{2}\)
These are the complex solutions to the quadratic equation \(20x^2 + 20x + 10 = 0\).
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