Question 311967
Put the equation in the form {{{ax^2+bx+c=0}}} first. 
{{{3 + 8z^2 = -7z}}}
{{{8z^2+7z+3=0}}}
Now comparing coefficients to the general equation,
{{{a=8}}}
{{{b=7}}}
{{{c=3}}}
The formula for the discriminant is,
{{{D=b^2-4ac=(7)^2-4(8)(3)=49-96=-47}}}
Since {{{D<0}}}, you will have two different imaginary roots. 
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Graphically, the curve never touches the x-axis, there are no real solutions. 
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{{{ graph( 300, 300, -5, 5, -5, 5, 8x^2+7x+3) }}} 
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Also, if {{{D=0}}}, you have 1 rational root, a double root.
If {{{D>0}}}, you have 2 distinct roots. If {{{D}}} is a perfect square(1,4,9,16,...), then the two roots will be rational, if not then they will be irrational.