4x^2=5x+1

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Solution for 4x^2=5x+1 equation:


Simplifying
4x2 = 5x + 1

Reorder the terms:
4x2 = 1 + 5x

Solving
4x2 = 1 + 5x

Solving for variable 'x'.

Reorder the terms:
-1 + -5x + 4x2 = 1 + 5x + -1 + -5x

Reorder the terms:
-1 + -5x + 4x2 = 1 + -1 + 5x + -5x

Combine like terms: 1 + -1 = 0
-1 + -5x + 4x2 = 0 + 5x + -5x
-1 + -5x + 4x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
-1 + -5x + 4x2 = 0

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
-0.25 + -1.25x + x2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -1.25x + 0.25 + x2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -1.25x + x2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -1.25x + x2 = 0 + 0.25
-1.25x + x2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-1.25x + x2 = 0.25

The x term is -1.25x.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25x + 0.390625 + x2 = 0.25 + 0.390625

Reorder the terms:
0.390625 + -1.25x + x2 = 0.25 + 0.390625

Combine like terms: 0.25 + 0.390625 = 0.640625
0.390625 + -1.25x + x2 = 0.640625

Factor a perfect square on the left side:
(x + -0.625)(x + -0.625) = 0.640625

Calculate the square root of the right side: 0.80039053

Break this problem into two subproblems by setting 
(x + -0.625) equal to 0.80039053 and -0.80039053.

Subproblem 1

x + -0.625 = 0.80039053 Simplifying x + -0.625 = 0.80039053 Reorder the terms: -0.625 + x = 0.80039053 Solving -0.625 + x = 0.80039053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + x = 0.80039053 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x = 0.80039053 + 0.625 x = 0.80039053 + 0.625 Combine like terms: 0.80039053 + 0.625 = 1.42539053 x = 1.42539053 Simplifying x = 1.42539053

Subproblem 2

x + -0.625 = -0.80039053 Simplifying x + -0.625 = -0.80039053 Reorder the terms: -0.625 + x = -0.80039053 Solving -0.625 + x = -0.80039053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + x = -0.80039053 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x = -0.80039053 + 0.625 x = -0.80039053 + 0.625 Combine like terms: -0.80039053 + 0.625 = -0.17539053 x = -0.17539053 Simplifying x = -0.17539053

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.42539053, -0.17539053}

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