How many solutions does #4x^2 + 8x + 3 = 0# have?
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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"):}#
