16x^2=40x-19

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Solution for 16x^2=40x-19 equation:


Simplifying
16x2 = 40x + -19

Reorder the terms:
16x2 = -19 + 40x

Solving
16x2 = -19 + 40x

Solving for variable 'x'.

Reorder the terms:
19 + -40x + 16x2 = -19 + 40x + 19 + -40x

Reorder the terms:
19 + -40x + 16x2 = -19 + 19 + 40x + -40x

Combine like terms: -19 + 19 = 0
19 + -40x + 16x2 = 0 + 40x + -40x
19 + -40x + 16x2 = 40x + -40x

Combine like terms: 40x + -40x = 0
19 + -40x + 16x2 = 0

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

Divide each side by '16'.
1.1875 + -2.5x + x2 = 0

Move the constant term to the right:

Add '-1.1875' to each side of the equation.
1.1875 + -2.5x + -1.1875 + x2 = 0 + -1.1875

Reorder the terms:
1.1875 + -1.1875 + -2.5x + x2 = 0 + -1.1875

Combine like terms: 1.1875 + -1.1875 = 0.0000
0.0000 + -2.5x + x2 = 0 + -1.1875
-2.5x + x2 = 0 + -1.1875

Combine like terms: 0 + -1.1875 = -1.1875
-2.5x + x2 = -1.1875

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 = -1.1875 + 1.5625

Reorder the terms:
1.5625 + -2.5x + x2 = -1.1875 + 1.5625

Combine like terms: -1.1875 + 1.5625 = 0.375
1.5625 + -2.5x + x2 = 0.375

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 0.375

Calculate the square root of the right side: 0.612372436

Break this problem into two subproblems by setting 
(x + -1.25) equal to 0.612372436 and -0.612372436.

Subproblem 1

x + -1.25 = 0.612372436 Simplifying x + -1.25 = 0.612372436 Reorder the terms: -1.25 + x = 0.612372436 Solving -1.25 + x = 0.612372436 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 = 0.612372436 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 0.612372436 + 1.25 x = 0.612372436 + 1.25 Combine like terms: 0.612372436 + 1.25 = 1.862372436 x = 1.862372436 Simplifying x = 1.862372436

Subproblem 2

x + -1.25 = -0.612372436 Simplifying x + -1.25 = -0.612372436 Reorder the terms: -1.25 + x = -0.612372436 Solving -1.25 + x = -0.612372436 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 = -0.612372436 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -0.612372436 + 1.25 x = -0.612372436 + 1.25 Combine like terms: -0.612372436 + 1.25 = 0.637627564 x = 0.637627564 Simplifying x = 0.637627564

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.862372436, 0.637627564}

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