x=4x^2+6x-16

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Solution for x=4x^2+6x-16 equation:


Simplifying
x = 4x2 + 6x + -16

Reorder the terms:
x = -16 + 6x + 4x2

Solving
x = -16 + 6x + 4x2

Solving for variable 'x'.

Reorder the terms:
16 + x + -6x + -4x2 = -16 + 6x + 4x2 + 16 + -6x + -4x2

Combine like terms: x + -6x = -5x
16 + -5x + -4x2 = -16 + 6x + 4x2 + 16 + -6x + -4x2

Reorder the terms:
16 + -5x + -4x2 = -16 + 16 + 6x + -6x + 4x2 + -4x2

Combine like terms: -16 + 16 = 0
16 + -5x + -4x2 = 0 + 6x + -6x + 4x2 + -4x2
16 + -5x + -4x2 = 6x + -6x + 4x2 + -4x2

Combine like terms: 6x + -6x = 0
16 + -5x + -4x2 = 0 + 4x2 + -4x2
16 + -5x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
16 + -5x + -4x2 = 0

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

Divide each side by '-4'.
-4 + 1.25x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 1.25x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + 1.25x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 1.25x + x2 = 0 + 4
1.25x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
1.25x + x2 = 4

The x term is 1.25x.  Take half its coefficient (0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
1.25x + 0.390625 + x2 = 4 + 0.390625

Reorder the terms:
0.390625 + 1.25x + x2 = 4 + 0.390625

Combine like terms: 4 + 0.390625 = 4.390625
0.390625 + 1.25x + x2 = 4.390625

Factor a perfect square on the left side:
(x + 0.625)(x + 0.625) = 4.390625

Calculate the square root of the right side: 2.095381827

Break this problem into two subproblems by setting 
(x + 0.625) equal to 2.095381827 and -2.095381827.

Subproblem 1

x + 0.625 = 2.095381827 Simplifying x + 0.625 = 2.095381827 Reorder the terms: 0.625 + x = 2.095381827 Solving 0.625 + x = 2.095381827 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = 2.095381827 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = 2.095381827 + -0.625 x = 2.095381827 + -0.625 Combine like terms: 2.095381827 + -0.625 = 1.470381827 x = 1.470381827 Simplifying x = 1.470381827

Subproblem 2

x + 0.625 = -2.095381827 Simplifying x + 0.625 = -2.095381827 Reorder the terms: 0.625 + x = -2.095381827 Solving 0.625 + x = -2.095381827 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = -2.095381827 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = -2.095381827 + -0.625 x = -2.095381827 + -0.625 Combine like terms: -2.095381827 + -0.625 = -2.720381827 x = -2.720381827 Simplifying x = -2.720381827

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.470381827, -2.720381827}

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