7x=4x^2-1

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


Simplifying
7x = 4x2 + -1

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

Solving
7x = -1 + 4x2

Solving for variable 'x'.

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

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

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

Combine like terms: 4x2 + -4x2 = 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 = 1.015625
0.765625 + -1.75x + x2 = 1.015625

Factor a perfect square on the left side:
(x + -0.875)(x + -0.875) = 1.015625

Calculate the square root of the right side: 1.007782219

Break this problem into two subproblems by setting 
(x + -0.875) equal to 1.007782219 and -1.007782219.

Subproblem 1

x + -0.875 = 1.007782219 Simplifying x + -0.875 = 1.007782219 Reorder the terms: -0.875 + x = 1.007782219 Solving -0.875 + x = 1.007782219 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 = 1.007782219 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = 1.007782219 + 0.875 x = 1.007782219 + 0.875 Combine like terms: 1.007782219 + 0.875 = 1.882782219 x = 1.882782219 Simplifying x = 1.882782219

Subproblem 2

x + -0.875 = -1.007782219 Simplifying x + -0.875 = -1.007782219 Reorder the terms: -0.875 + x = -1.007782219 Solving -0.875 + x = -1.007782219 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 = -1.007782219 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = -1.007782219 + 0.875 x = -1.007782219 + 0.875 Combine like terms: -1.007782219 + 0.875 = -0.132782219 x = -0.132782219 Simplifying x = -0.132782219

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.882782219, -0.132782219}

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