SOLUTION: 1. Determine whether the following equations have a solution or not? Justify your answer. a) x2 + 3x - 15 = 0 b) x2 + x + 4 = 0 c) x2 – 4x - 7 = 0 d) x2 – 8x + 16 = 0

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Question 181251: 1. Determine whether the following equations have a solution or not? Justify your answer.
a) x2 + 3x - 15 = 0
b) x2 + x + 4 = 0
c) x2 – 4x - 7 = 0
d) x2 – 8x + 16 = 0
e) 2x2 - 3x + 7 = 0
f) x2 – 4x - 77 = 0
g) 3x2 - 7x + 6 = 0
h) 4x2 + 16x + 16 = 0
I do not understand this one. Please help.
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Determine whether the following equations have a solution or not? Justify your answer.
a) x2 + 3x - 15 = 0
b) x2 + x + 4 = 0
c) x2 – 4x - 7 = 0
d) x2 – 8x + 16 = 0
e) 2x2 - 3x + 7 = 0
f) x2 – 4x - 77 = 0
g) 3x2 - 7x + 6 = 0
h) 4x2 + 16x + 16 = 0
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All quadratics have solutions. The solution might not be real numbers, but they're still solutions.
For example:
a) x2 + 3x - 15 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B3x%2B-15+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%283%29%5E2-4%2A1%2A-15=69.

Discriminant d=69 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-3%2B-sqrt%28+69+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%283%29%2Bsqrt%28+69+%29%29%2F2%5C1+=+2.65331193145904
x%5B2%5D+=+%28-%283%29-sqrt%28+69+%29%29%2F2%5C1+=+-5.65331193145904

Quadratic expression 1x%5E2%2B3x%2B-15 can be factored:
1x%5E2%2B3x%2B-15+=+%28x-2.65331193145904%29%2A%28x--5.65331193145904%29
Again, the answer is: 2.65331193145904, -5.65331193145904. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B3%2Ax%2B-15+%29

This has real number solutions, tho the numbers are irrational.
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b) x2 + x + 4 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B1x%2B4+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%281%29%5E2-4%2A1%2A4=-15.

The discriminant -15 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.


In the field of imaginary numbers, the square root of -15 is + or - sqrt%28+15%29+=+3.87298334620742.

The solution is , or
Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B1%2Ax%2B4+%29

The solutions are complex numbers, but they are solutions.