Question - Solving a Complex Equation with Imaginary Numbers
Solution:
The image displays a mathematical problem:\[ \text{Find the value of a and b if } bi^{2} + \frac{4}{3}i - \sqrt{3} = a + bi\sqrt{3}. \]To solve this, we need to realize that $$i^2 = -1$$ and group real and imaginary parts separately.\[ bi^{2} + \frac{4}{3}i - \sqrt{3} = a + bi\sqrt{3}, \]\[ b(-1) + \frac{4}{3}i - \sqrt{3} = a + bi\sqrt{3}. \]Now, group real parts together and imaginary parts together:Real parts: $$-b - \sqrt{3} = a,$$Imaginary parts: $$\frac{4}{3}i = bi\sqrt{3}.$$Now let's isolate $$b$$ from the imaginary part:\[ b = \frac{\frac{4}{3}}{\sqrt{3}} = \frac{4}{3\sqrt{3}} = \frac{4}{3}\cdot\frac{\sqrt{3}}{3} = \frac{4\sqrt{3}}{9}.\]Then substitute $$b$$ into the real part to find $$a$$:\[ -\left(\frac{4\sqrt{3}}{9}\right) - \sqrt{3} = a,\]\[ a = -\left(\frac{4\sqrt{3}}{9} + \frac{9\sqrt{3}}{9}\right), \]\[ a = -\left(\frac{4\sqrt{3} + 9\sqrt{3}}{9}\right), \]\[ a = -\left(\frac{13\sqrt{3}}{9}\right). \]Thus, the values for $$a$$ and $$b$$ are:\[ a = -\frac{13\sqrt{3}}{9}, \]\[ b = \frac{4\sqrt{3}}{9}. \]
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