5x^2=-3x+4

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Solution for 5x^2=-3x+4 equation:


Simplifying
5x2 = -3x + 4

Reorder the terms:
5x2 = 4 + -3x

Solving
5x2 = 4 + -3x

Solving for variable 'x'.

Reorder the terms:
-4 + 3x + 5x2 = 4 + -3x + -4 + 3x

Reorder the terms:
-4 + 3x + 5x2 = 4 + -4 + -3x + 3x

Combine like terms: 4 + -4 = 0
-4 + 3x + 5x2 = 0 + -3x + 3x
-4 + 3x + 5x2 = -3x + 3x

Combine like terms: -3x + 3x = 0
-4 + 3x + 5x2 = 0

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

Divide each side by '5'.
-0.8 + 0.6x + x2 = 0

Move the constant term to the right:

Add '0.8' to each side of the equation.
-0.8 + 0.6x + 0.8 + x2 = 0 + 0.8

Reorder the terms:
-0.8 + 0.8 + 0.6x + x2 = 0 + 0.8

Combine like terms: -0.8 + 0.8 = 0.0
0.0 + 0.6x + x2 = 0 + 0.8
0.6x + x2 = 0 + 0.8

Combine like terms: 0 + 0.8 = 0.8
0.6x + x2 = 0.8

The x term is 0.6x.  Take half its coefficient (0.3).
Square it (0.09) and add it to both sides.

Add '0.09' to each side of the equation.
0.6x + 0.09 + x2 = 0.8 + 0.09

Reorder the terms:
0.09 + 0.6x + x2 = 0.8 + 0.09

Combine like terms: 0.8 + 0.09 = 0.89
0.09 + 0.6x + x2 = 0.89

Factor a perfect square on the left side:
(x + 0.3)(x + 0.3) = 0.89

Calculate the square root of the right side: 0.943398113

Break this problem into two subproblems by setting 
(x + 0.3) equal to 0.943398113 and -0.943398113.

Subproblem 1

x + 0.3 = 0.943398113 Simplifying x + 0.3 = 0.943398113 Reorder the terms: 0.3 + x = 0.943398113 Solving 0.3 + x = 0.943398113 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + x = 0.943398113 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + x = 0.943398113 + -0.3 x = 0.943398113 + -0.3 Combine like terms: 0.943398113 + -0.3 = 0.643398113 x = 0.643398113 Simplifying x = 0.643398113

Subproblem 2

x + 0.3 = -0.943398113 Simplifying x + 0.3 = -0.943398113 Reorder the terms: 0.3 + x = -0.943398113 Solving 0.3 + x = -0.943398113 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + x = -0.943398113 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + x = -0.943398113 + -0.3 x = -0.943398113 + -0.3 Combine like terms: -0.943398113 + -0.3 = -1.243398113 x = -1.243398113 Simplifying x = -1.243398113

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.643398113, -1.243398113}

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