Question

What is the answer to this equation using the quadratic formula? 3x^2+2x+4x=0

Answer 1

Answer: No real number solutions

The complex number solutions are
`x = (-1+ i√(11))/(3) \ \text{ or } \ x = (-1- i√(11))/(3)\\\\`

where
`i = √(-1)`

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Step-by-step explanation:

The given equation 3x^2+2x+4 = 0 is in the form ax^2 + bx + c = 0

We have

• a = 3
• b = 2
• c = 4

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

`x = (-b\pm√(b^2-4ac))/(2a)\\\\x = (-(2)\pm√((2)^2-4(3)(4)))/(2(3))\\\\x = (-2\pm√(-44))/(6)\\\\x = (-2\pm√(-1*4*11))/(6)\\\\x = (-2\pm√(-1)*√(4)*√(11))/(6)\\\\`

`x = (-2\pm i*2*√(11))/(6)\\\\x = (-2\pm2i√(11))/(6)\\\\x = (2(-1\pm i√(11)))/(6)\\\\x = (-1\pm i√(11))/(3)\\\\x = (-1+ i√(11))/(3) \ \text{ or } \ x = (-1- i√(11))/(3)\\\\`

Notice in step 3 we have -44 under the square root. The negative value in the square root leads directly to the imaginary number
`i = √(-1)`

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).