Evaluate (-12+ square root of -45)/24

Problem

$\frac{-12+\sqrt{-45}}{24}$

Solution

1. Identify the square root of a negative number as an imaginary number using the property $\sqrt{-a}=i\sqrt{a}$

$\sqrt{-45}=i\sqrt{45}$

2. Simplify the radical by factoring out the largest perfect square, which is $9$

$\sqrt{45}=\sqrt{9\cdot 5}=3\sqrt{5}$

3. Substitute the simplified radical back into the original expression.

$\frac{-12+3i\sqrt{5}}{24}$

4. Factor out the greatest common factor from the numerator, which is $3$

$3(-4+i\sqrt{5})$

5. Divide both the numerator and the denominator by $3$ to simplify the fraction.

$\frac{3(-4+i\sqrt{5})}{24}=\frac{-4+i\sqrt{5}}{8}$

6. Rewrite the expression in standard complex form $a+bi$ by splitting the fraction.

$\frac{-4}{8}+\frac{i\sqrt{5}}{8}$

7. Reduce the real part to its simplest form.

$-\frac{1}{2}+\frac{\sqrt{5}}{8}i$

Final Answer

$\frac{-12+\sqrt{-45}}{24}=-\frac{1}{2}+\frac{\sqrt{5}}{8}i$

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