Question

How many solutions does `4x^2 + 8x + 3 = 0` have?

Answer 1

The easiest way to determine the answer to your question is to evaluate the `color(red)("discriminant")`
For a quadratic in the form:
`y = ax^2+bx+c`
the `color(red)("discriminant"`) is equal to
`color(red)(b^2-4ac)`

For your example
`4x^2+8x+3=0`
The `color(red)("discriminant")` is
`Delta = (8^2-4(4)(3))/(2(4))`

`Delta=2`

The number of solutions is determined by the value of the `color(red)("discriminant, " Delta)`

`Delta { (>0 " two solutions"),(=0" one solution"),(<0" no solutions"):}`

So
`4x^2+8x+3=0` has two solutions

Further explanation:
The discriminant comes from the formula for quadratic root solutions
`(-b+-sqrt(color(red)(b^2-4ac)))/(2a)`
and it should be fairly obvious from this to understand why

`color(red)(b^2-4ac) { (>0 rarr" two solutions"),(=0 rarr" one solution"),(<0 rarr " no solutions"):}`