Answer: No real number solutions
The complex number solutions are [tex]x = \frac{-1+ i\sqrt{11}}{3} \ \text{ or } \ x = \frac{-1- i\sqrt{11}}{3}\\\\[/tex]
where [tex]i = \sqrt{-1}[/tex]
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Explanation:
The given equation 3x^2+2x+4 = 0 is in the form ax^2 + bx + c = 0
We have
Let's compute the discriminant.
d = b^2 - 4ac
d = (2)^2-4(3)(4)
d = -44
The result is negative, so there are no real number solutions.
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If you wanted to find the complex valued solutions, then we apply the quadratic formula.
Plug in a = 3, b = 2, c = 4
[tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-(2)\pm\sqrt{(2)^2-4(3)(4)}}{2(3)}\\\\x = \frac{-2\pm\sqrt{-44}}{6}\\\\x = \frac{-2\pm\sqrt{-1*4*11}}{6}\\\\x = \frac{-2\pm\sqrt{-1}*\sqrt{4}*\sqrt{11}}{6}\\\\[/tex]
[tex]x = \frac{-2\pm i*2*\sqrt{11}}{6}\\\\x = \frac{-2\pm2i\sqrt{11}}{6}\\\\x = \frac{2(-1\pm i\sqrt{11})}{6}\\\\x = \frac{-1\pm i\sqrt{11}}{3}\\\\x = \frac{-1+ i\sqrt{11}}{3} \ \text{ or } \ x = \frac{-1- i\sqrt{11}}{3}\\\\[/tex]
Notice in step 3 we have -44 under the square root. The negative value in the square root leads directly to the imaginary number [tex]i = \sqrt{-1}[/tex]
Though the term "imaginary" is a bit unfair and misleading because numbers like -22 are just as imaginary and made up by humans. It just depends on context in which imaginary numbers are useful (eg: with physics or engineering).
