0=-5x^2+4x+5

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Solution for 0=-5x^2+4x+5 equation:


Simplifying
0 = -5x2 + 4x + 5

Reorder the terms:
0 = 5 + 4x + -5x2

Solving
0 = 5 + 4x + -5x2

Solving for variable 'x'.

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

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

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

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

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

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

Divide each side by '5'.
-1 + -0.8x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.8x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.8x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.8x + x2 = 0 + 1
-0.8x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.8x + x2 = 1

The x term is -0.8x.  Take half its coefficient (-0.4).
Square it (0.16) and add it to both sides.

Add '0.16' to each side of the equation.
-0.8x + 0.16 + x2 = 1 + 0.16

Reorder the terms:
0.16 + -0.8x + x2 = 1 + 0.16

Combine like terms: 1 + 0.16 = 1.16
0.16 + -0.8x + x2 = 1.16

Factor a perfect square on the left side:
(x + -0.4)(x + -0.4) = 1.16

Calculate the square root of the right side: 1.077032961

Break this problem into two subproblems by setting 
(x + -0.4) equal to 1.077032961 and -1.077032961.

Subproblem 1

x + -0.4 = 1.077032961 Simplifying x + -0.4 = 1.077032961 Reorder the terms: -0.4 + x = 1.077032961 Solving -0.4 + x = 1.077032961 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + x = 1.077032961 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + x = 1.077032961 + 0.4 x = 1.077032961 + 0.4 Combine like terms: 1.077032961 + 0.4 = 1.477032961 x = 1.477032961 Simplifying x = 1.477032961

Subproblem 2

x + -0.4 = -1.077032961 Simplifying x + -0.4 = -1.077032961 Reorder the terms: -0.4 + x = -1.077032961 Solving -0.4 + x = -1.077032961 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + x = -1.077032961 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + x = -1.077032961 + 0.4 x = -1.077032961 + 0.4 Combine like terms: -1.077032961 + 0.4 = -0.677032961 x = -0.677032961 Simplifying x = -0.677032961

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.477032961, -0.677032961}

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