5+23x-16x^2=13

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Solution for 5+23x-16x^2=13 equation:


Simplifying
5 + 23x + -16x2 = 13

Solving
5 + 23x + -16x2 = 13

Solving for variable 'x'.

Reorder the terms:
5 + -13 + 23x + -16x2 = 13 + -13

Combine like terms: 5 + -13 = -8
-8 + 23x + -16x2 = 13 + -13

Combine like terms: 13 + -13 = 0
-8 + 23x + -16x2 = 0

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

Divide each side by '-16'.
0.5 + -1.4375x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -1.4375x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -1.4375x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -1.4375x + x2 = 0 + -0.5
-1.4375x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-1.4375x + x2 = -0.5

The x term is -1.4375x.  Take half its coefficient (-0.71875).
Square it (0.5166015625) and add it to both sides.

Add '0.5166015625' to each side of the equation.
-1.4375x + 0.5166015625 + x2 = -0.5 + 0.5166015625

Reorder the terms:
0.5166015625 + -1.4375x + x2 = -0.5 + 0.5166015625

Combine like terms: -0.5 + 0.5166015625 = 0.0166015625
0.5166015625 + -1.4375x + x2 = 0.0166015625

Factor a perfect square on the left side:
(x + -0.71875)(x + -0.71875) = 0.0166015625

Calculate the square root of the right side: 0.128847051

Break this problem into two subproblems by setting 
(x + -0.71875) equal to 0.128847051 and -0.128847051.

Subproblem 1

x + -0.71875 = 0.128847051 Simplifying x + -0.71875 = 0.128847051 Reorder the terms: -0.71875 + x = 0.128847051 Solving -0.71875 + x = 0.128847051 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.71875' to each side of the equation. -0.71875 + 0.71875 + x = 0.128847051 + 0.71875 Combine like terms: -0.71875 + 0.71875 = 0.00000 0.00000 + x = 0.128847051 + 0.71875 x = 0.128847051 + 0.71875 Combine like terms: 0.128847051 + 0.71875 = 0.847597051 x = 0.847597051 Simplifying x = 0.847597051

Subproblem 2

x + -0.71875 = -0.128847051 Simplifying x + -0.71875 = -0.128847051 Reorder the terms: -0.71875 + x = -0.128847051 Solving -0.71875 + x = -0.128847051 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.71875' to each side of the equation. -0.71875 + 0.71875 + x = -0.128847051 + 0.71875 Combine like terms: -0.71875 + 0.71875 = 0.00000 0.00000 + x = -0.128847051 + 0.71875 x = -0.128847051 + 0.71875 Combine like terms: -0.128847051 + 0.71875 = 0.589902949 x = 0.589902949 Simplifying x = 0.589902949

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.847597051, 0.589902949}

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