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.

How do you use the discriminant to determine the nature of the roots for 4x^2 + 15x + 10 = 04x^2 + 15x + 10 = 0?