Question - Solving Quadratic Equations with Complex Solutions
Solution:
The equation given in the image is a quadratic equation of the form:\[ 2x^2 - 4x + 4 = 0 \]We can solve this equation by either factoring, completing the square, or using the quadratic formula. This equation does not factor easily, so we'll use the quadratic formula:\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]For the given equation, $$a = 2$$, $$b = -4$$, and $$c = 4$$. Plugging these values into the quadratic formula:\[ x = \frac{{-(-4) \pm \sqrt{{(-4)^2 - 4(2)(4)}}}}{{2(2)}} \]\[ x = \frac{{4 \pm \sqrt{{16 - 32}}}}{{4}} \]\[ x = \frac{{4 \pm \sqrt{{-16}}}}{{4}} \]Since we have a negative number under the square root, the solutions will be complex numbers. We continue:\[ x = \frac{{4 \pm 4i}}{{4}} \]Simplifying further, we divide both terms in the numerator by the denominator:\[ x = 1 \pm i \]The solutions to the quadratic equation are:\[ x = 1 + i \quad \text{and} \quad x = 1 - i \]Here, $$i$$ is the imaginary unit, where $$i^2 = -1$$.
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