Question 177858
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The discriminant is that part of the quadratic equation under the radical, so *[tex \Large \Delta = b^2 - 4ac]


If *[tex \Large \Delta > 0] then the quadratic equation has two real and unequal roots.


If *[tex \Large \Delta = 0] then the quadratic equation has two real and equal roots.  It is also said that the quadratic has one real root with a multiplicity of 2.  This is because the *[tex \Large \Delta = 0] situation only occurs when the quadratic is a perfect square as in *[tex \Large x^2 - 2ax + a^2 = (x - a)(x - a) = 0].  Each one of the factors translates to a root of the equation, so there are, in fact, two of them; they just happen to be equal.


If *[tex \Large \Delta < 0] then the quadratic equation has a conjugate pair of complex roots of the form *[tex \Large a \pm bi] where <i>i</i> is the imaginary number defined by *[tex \Large i^2 = -1].  Note that unless the real part of the complex number (the <i>a</i> in  *[tex \Large a \pm bi]) is zero, the roots are not purely imaginary.


So, for each of your problems, calculate *[tex \Large \Delta ] and evaluate the character of the roots per the definitions above.


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