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

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Question 170775This question is from textbook
: What type of solution do you get for quadratic equations where D<0?
Give reasons for your answer.
Also provide an example of such a quadratic equation and find the solution of the equation.
I have tried three times and still I am wrong, please help. This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If D%3C0, then the quadratic will have two complex (ie non real) solutions.


Why? Recall that the quadratic formula is

x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29+


The discriminant D is everything in the square root. So this means that

D=b%5E2-4ac


Now if D%3C0, this means that "D" is negative. Remember you CANNOT take the square root of a negative number. This is why you will NOT get any real solutions if D%3C0


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For example, let's find the discriminant for y=x%5E2%2B2x%2B5


From x%5E2%2B2x%2B5 we can see that a=1, b=2, and c=5


D=b%5E2-4ac Start with the discriminant formula


D=%282%29%5E2-4%281%29%285%29 Plug in a=1, b=2, and c=5


D=4-4%281%29%285%29 Square 2 to get 4


D=4-20 Multiply 4%281%29%285%29 to get %284%29%285%29=20


D=-16 Subtract 20 from 4 to get -16


Since the discriminant is less than zero, this means that there are two complex solutions



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Now let's use the quadratic formula to find the solutions of y=x%5E2%2B2x%2B5


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%282%29+%2B-+sqrt%28+%282%29%5E2-4%281%29%285%29+%29%29%2F%282%281%29%29 Plug in a=1, b=2, and c=5


x+=+%28-2+%2B-+sqrt%28+4-4%281%29%285%29+%29%29%2F%282%281%29%29 Square 2 to get 4.


x+=+%28-2+%2B-+sqrt%28+4-20+%29%29%2F%282%281%29%29 Multiply 4%281%29%285%29 to get 20


x+=+%28-2+%2B-+sqrt%28+-16+%29%29%2F%282%281%29%29 Subtract 20 from 4 to get -16


x+=+%28-2+%2B-+sqrt%28+-16+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%28-2+%2B-+4%2Ai%29%2F%282%29 Simplify sqrt%28-16%29 to get sqrt%28-16%29=sqrt%2816%28-1%29%29=sqrt%2816%29%2Asqrt%28-1%29=4%2Ai. Note: i=sqrt%28-1%29


x+=+%28-2+%2B+4%2Ai%29%2F%282%29 or x+=+%28-2+-+4%2Ai%29%2F%282%29 Break up the expression.


x+=+%28-2%29%2F%282%29+%2B+%284%2Ai%29%2F%282%29 or x+=++%28-2%29%2F%282%29+-+%284%2Ai%29%2F%282%29 Break up the fraction for each case.


x+=+-1%2B2%2Ai or x+=++-1-2%2Ai Reduce.


So our answers are x+=+-1%2B2%2Ai or x+=+-1-2%2Ai


Since our answers are complex (ie not real), this verifies our original claim.