Solving a Complex Equation with Imaginary Numbers
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:
\[ a = -\frac{13\sqrt{3}}{9}, \]
\[ b = \frac{4\sqrt{3}}{9}. \]
