10x=4x^2-1

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Solution for 10x=4x^2-1 equation:


Simplifying
10x = 4x2 + -1

Reorder the terms:
10x = -1 + 4x2

Solving
10x = -1 + 4x2

Solving for variable 'x'.

Reorder the terms:
1 + 10x + -4x2 = -1 + 4x2 + 1 + -4x2

Reorder the terms:
1 + 10x + -4x2 = -1 + 1 + 4x2 + -4x2

Combine like terms: -1 + 1 = 0
1 + 10x + -4x2 = 0 + 4x2 + -4x2
1 + 10x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
1 + 10x + -4x2 = 0

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

Divide each side by '-4'.
-0.25 + -2.5x + x2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -2.5x + 0.25 + x2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -2.5x + x2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -2.5x + x2 = 0 + 0.25
-2.5x + x2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-2.5x + x2 = 0.25

The x term is -2.5x.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x + 1.5625 + x2 = 0.25 + 1.5625

Reorder the terms:
1.5625 + -2.5x + x2 = 0.25 + 1.5625

Combine like terms: 0.25 + 1.5625 = 1.8125
1.5625 + -2.5x + x2 = 1.8125

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 1.8125

Calculate the square root of the right side: 1.346291202

Break this problem into two subproblems by setting 
(x + -1.25) equal to 1.346291202 and -1.346291202.

Subproblem 1

x + -1.25 = 1.346291202 Simplifying x + -1.25 = 1.346291202 Reorder the terms: -1.25 + x = 1.346291202 Solving -1.25 + x = 1.346291202 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 1.346291202 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 1.346291202 + 1.25 x = 1.346291202 + 1.25 Combine like terms: 1.346291202 + 1.25 = 2.596291202 x = 2.596291202 Simplifying x = 2.596291202

Subproblem 2

x + -1.25 = -1.346291202 Simplifying x + -1.25 = -1.346291202 Reorder the terms: -1.25 + x = -1.346291202 Solving -1.25 + x = -1.346291202 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -1.346291202 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -1.346291202 + 1.25 x = -1.346291202 + 1.25 Combine like terms: -1.346291202 + 1.25 = -0.096291202 x = -0.096291202 Simplifying x = -0.096291202

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.596291202, -0.096291202}

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