SOLUTION: What type of solution do you get for quadratic equations where D < 0? Give reason for your answer. Provide an example of such a quadratic equation and find the solution of the equa

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Question 113339: What type of solution do you get for quadratic equations where D < 0? Give reason for your answer. Provide an example of such a quadratic equation and find the solution of the equation.
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
By D, I presume you mean the discriminant b%5E2-4ac part of the quadratic formula.

When D < 0, the solution to the equation will be a conjugate pair of complex number roots. This is because sqrt%28-1%29 is not defined in the real number system so we have to use the imaginary number i where i%5E2=-1 to express the roots of the equation. This conjugate pair of complex numbers is always in the form of a%2B-bi, though it is possible for the real number part of the complex number to be zero, as in the solution to x%5E2%2B5=0 (The solutions are i%2Asqrt%285%29 and -i%2Asqrt%285%29 which are equivalent to 0%2Bi%2Asqrt%285%29 and 0-i%2Asqrt%285%29. Considering the function f%28x%29=ax%5E2%2Bbx%2Bc, if b%5E2-4ac%3C0, then the graph of the function will not intersect the x-axis.

An example of such an equation is x%5E2%2B2x%2B5. Here, the discriminant is 4-20=-16, and the solution set would contain a conjugate pair of complex roots given by:

x=%28-2%2B-i%2Asqrt%2816%29%29%2F2
x=-1%2B2i or x=-1-2i

graph%28400%2C400%2C-5%2C5%2C-2%2C10%2Cx%5E2%2B2x%2B5%29

Notice that the graph does NOT intersect the x-axis.

Hope this helps.
John
P.S. Super-Double-Plus Extra Credit. Now that we know that there are ALWAYS two roots to a second degree or quadratic equation, how many roots would you say a third degree or cubic equation ALWAYS has?