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The solutions to $4x^2 + 3 = 3x - 9$ can be written in the form $x = a \pm b i,$ where $a$ and $b$ are real numbers. What is $a + b^2$? Express your answer as a fraction.

 Jun 25, 2019
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The solutions to  \(4x^2 + 3 = 3x - 9\)  can be written in the form  \(x = a \pm b i,\)  where  \(a\)  and  \(b\)  are real numbers. What is  \(a + b^2\) ?  Express your answer as a fraction.

 

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\(4x^2 + 3 = 3x - 9\)

                                        Subtract  3x  from both sides and add  9  to both sides of the equation.

\(4x^2-3x + 12 = 0\)

                                        Now we can use the quadratic formula to solve for  x

 

\(x\ = \ \frac{3\pm\sqrt{3^2-4(4)(12)}}{2(4)}\ =\ \frac{3\pm\sqrt{-183}}{8}\ =\ \frac{3\pm\sqrt{183}i}{8}\ =\ \frac38\pm\frac{\sqrt{183}}{8}i\)

 

And now we have the solutions in the form   \(x = a \pm b i\)   so we can see...

 

\(a+b^2\ =\ \Big(\frac38\Big)+\Big(\frac{\sqrt{183}}{8}\Big)^2\ =\ \frac38+\frac{183}{64}\ =\ \frac{24}{64}+\frac{183}{64}\ =\ \frac{207}{64}\)_

 Jun 25, 2019

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