How do you use the discriminant to determine the nature of the roots for #4x^2 + 15x + 10 = 0#?

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Answer 1 · 2015-06-27T22:01:39

#4x^2+15x+10# has discriminant #Delta = 65#

So #4x^2+15x+10 = 0# has two distinct real, irrational roots.

#4x^2+15x+10# is of the form #ax^2+bx+c#
with #a=4#, #b=15# and #c=10#.

The discriminant is given by the formula:

#Delta = b^2 - 4ac = 15^2 - (4 xx 4 xx 10) = 225 - 160 = 65#

This is positive, but not a perfect square. So the quadratic equation has two distinct irrational real roots.

The various possible cases (assuming that the quadratic has rational coefficients) are as follows:

#Delta > 0# The equation has two distinct real roots. If #Delta# is a perfect square then the roots are rational too. Otherwise they are irrational.

#Delta = 0# The equation has one (repeated) rational real root.

#Delta < 0# The equation has no real roots. It has two distinct complex roots, which are complex conjugates of one another.