5x^2=3x+5

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


Simplifying
5x2 = 3x + 5

Reorder the terms:
5x2 = 5 + 3x

Solving
5x2 = 5 + 3x

Solving for variable 'x'.

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

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

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

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

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

Divide each side by '5'.
-1 + -0.6x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.6x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.6x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.6x + x2 = 0 + 1
-0.6x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.6x + x2 = 1

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 = 1 + 0.09

Reorder the terms:
0.09 + -0.6x + x2 = 1 + 0.09

Combine like terms: 1 + 0.09 = 1.09
0.09 + -0.6x + x2 = 1.09

Factor a perfect square on the left side:
(x + -0.3)(x + -0.3) = 1.09

Calculate the square root of the right side: 1.044030651

Break this problem into two subproblems by setting 
(x + -0.3) equal to 1.044030651 and -1.044030651.

Subproblem 1

x + -0.3 = 1.044030651 Simplifying x + -0.3 = 1.044030651 Reorder the terms: -0.3 + x = 1.044030651 Solving -0.3 + x = 1.044030651 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 = 1.044030651 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + x = 1.044030651 + 0.3 x = 1.044030651 + 0.3 Combine like terms: 1.044030651 + 0.3 = 1.344030651 x = 1.344030651 Simplifying x = 1.344030651

Subproblem 2

x + -0.3 = -1.044030651 Simplifying x + -0.3 = -1.044030651 Reorder the terms: -0.3 + x = -1.044030651 Solving -0.3 + x = -1.044030651 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 = -1.044030651 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + x = -1.044030651 + 0.3 x = -1.044030651 + 0.3 Combine like terms: -1.044030651 + 0.3 = -0.744030651 x = -0.744030651 Simplifying x = -0.744030651

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.344030651, -0.744030651}

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