4x^2+1=7x

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Solution for 4x^2+1=7x equation:


Simplifying
4x2 + 1 = 7x

Reorder the terms:
1 + 4x2 = 7x

Solving
1 + 4x2 = 7x

Solving for variable 'x'.

Reorder the terms:
1 + -7x + 4x2 = 7x + -7x

Combine like terms: 7x + -7x = 0
1 + -7x + 4x2 = 0

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

Divide each side by '4'.
0.25 + -1.75x + x2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -1.75x + -0.25 + x2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -1.75x + x2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -1.75x + x2 = 0 + -0.25
-1.75x + x2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-1.75x + x2 = -0.25

The x term is -1.75x.  Take half its coefficient (-0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
-1.75x + 0.765625 + x2 = -0.25 + 0.765625

Reorder the terms:
0.765625 + -1.75x + x2 = -0.25 + 0.765625

Combine like terms: -0.25 + 0.765625 = 0.515625
0.765625 + -1.75x + x2 = 0.515625

Factor a perfect square on the left side:
(x + -0.875)(x + -0.875) = 0.515625

Calculate the square root of the right side: 0.718070331

Break this problem into two subproblems by setting 
(x + -0.875) equal to 0.718070331 and -0.718070331.

Subproblem 1

x + -0.875 = 0.718070331 Simplifying x + -0.875 = 0.718070331 Reorder the terms: -0.875 + x = 0.718070331 Solving -0.875 + x = 0.718070331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x = 0.718070331 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = 0.718070331 + 0.875 x = 0.718070331 + 0.875 Combine like terms: 0.718070331 + 0.875 = 1.593070331 x = 1.593070331 Simplifying x = 1.593070331

Subproblem 2

x + -0.875 = -0.718070331 Simplifying x + -0.875 = -0.718070331 Reorder the terms: -0.875 + x = -0.718070331 Solving -0.875 + x = -0.718070331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x = -0.718070331 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = -0.718070331 + 0.875 x = -0.718070331 + 0.875 Combine like terms: -0.718070331 + 0.875 = 0.156929669 x = 0.156929669 Simplifying x = 0.156929669

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.593070331, 0.156929669}

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