Question 329607: I really need help with this quadratic equation and check. Thank you for all your help.
1/4x² + 17/4 = 2x
Found 2 solutions by Fombitz, Theo:
Answer by Fombitz(32388)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! I believe you are saying:
(1/4)*x^2 + (17/4) = 2*x
If this is not the case, then send me an email with a clarification.
You can start by subtracting 2*x from both sides of the equation to get:
(1/4)*x^2 - 2*x + (17/4) = 0
Multiply both sides of this equation by 4 to get:
x^2 - 8*x + 17 = 0
Complete the squares on the (x^2 - 8*x) part to get:
(x-4)^2 - 16 + 17 = 0
Add 16 and subtract 17 from both sides of this equation to get:
(x-4)^2 = -1
Take the square root of both sides of this equation to get x-4 = +/- square root of (-1).
Since square root of (-1) is not a real value, then this quadratic equation does not have any real roots.
It has imaginary, or complex roots, complex being a combination of real and imaginary.
A graph of your original equation is shown below:
You can see that the graph of this equation does not cross the x-axis, therefore this equation doesn't have any real roots.
The roots that we did find are:
x = 4 +/- square root of (-1)
Since square root of (-1) is equal to i, then the roots become:
x = 4 +/- i.
Once again, these are not real roots.
These are complex roots, because they are a combination of a real part (the 4) and an imaginary part (the i, where i = square root of (-1)).
If you had used the quadratic formula, you would have gotten the same answer.
The quadratic formula is x = (-b +/- square root of (b^2-4ac) / (2a).
In your equation of x^2 - 8*x + 17 = 0,
a = 1
b = -8
c = 17
The quadratic formula becomes:
x = (-(-8) +/- square root of (64-4*1*17)) / (2*1) which becomes:
x = (8 +/- square root of (64 - 68)) / 2 which becomes:
x = (8 +/- square root of (-4) / 2 which becomes:
x = (8 +/- square root of (4*-1) / 2 which becomes:
x = (8 +/- 2 * square root of (-1) / 2 which becomes:
x = 4 +/- square root of (-1).
Since square root of (-1) is equal to i, then the solution becomes
x = 4 +/- i.
This is the same answer we derived using the completing the squares method.
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