Find the exact solution simplify as much as possible and show work

5x

^{2}+1=4x-
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Find the exact solution simplify as much as possible and show work

5x^{2}+1=4x

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5x^{2} + 1 = 4x

First, put the equation into standard form:

5x^{2} - 4x + 1 = 0

This equation's roots are complex numbers; it has no roots in the set of real numbers. To show this, compute the coordinates of the parabola's vertex. The x-coordinate is located at x = -b/2a (where a=5 and b=-4). To find the y-coordinate, plug the value of the x coordinate of the vertex into your quadratic equation.

Since the coefficient of the x^{2} term is positive (5), the parabola opens upward. If the y-coordinate of the vertex is positive, then the parabola will not cross the x-axis and will have no real roots. (The roots are the points where the graph crosses the x-axis)

To find the complex roots, use the quadratic formula:

x = (-b/2a) ± (1/2a)√(b^{2}-4ac)

For this problem, a = 5, b = -4, c = 1. Plug those values into the quadratic formula to get the answer.

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