5x^2=7x+3

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Solution for 5x^2=7x+3 equation:


Simplifying
5x2 = 7x + 3

Reorder the terms:
5x2 = 3 + 7x

Solving
5x2 = 3 + 7x

Solving for variable 'x'.

Reorder the terms:
-3 + -7x + 5x2 = 3 + 7x + -3 + -7x

Reorder the terms:
-3 + -7x + 5x2 = 3 + -3 + 7x + -7x

Combine like terms: 3 + -3 = 0
-3 + -7x + 5x2 = 0 + 7x + -7x
-3 + -7x + 5x2 = 7x + -7x

Combine like terms: 7x + -7x = 0
-3 + -7x + 5x2 = 0

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

Divide each side by '5'.
-0.6 + -1.4x + x2 = 0

Move the constant term to the right:

Add '0.6' to each side of the equation.
-0.6 + -1.4x + 0.6 + x2 = 0 + 0.6

Reorder the terms:
-0.6 + 0.6 + -1.4x + x2 = 0 + 0.6

Combine like terms: -0.6 + 0.6 = 0.0
0.0 + -1.4x + x2 = 0 + 0.6
-1.4x + x2 = 0 + 0.6

Combine like terms: 0 + 0.6 = 0.6
-1.4x + x2 = 0.6

The x term is -1.4x.  Take half its coefficient (-0.7).
Square it (0.49) and add it to both sides.

Add '0.49' to each side of the equation.
-1.4x + 0.49 + x2 = 0.6 + 0.49

Reorder the terms:
0.49 + -1.4x + x2 = 0.6 + 0.49

Combine like terms: 0.6 + 0.49 = 1.09
0.49 + -1.4x + x2 = 1.09

Factor a perfect square on the left side:
(x + -0.7)(x + -0.7) = 1.09

Calculate the square root of the right side: 1.044030651

Break this problem into two subproblems by setting 
(x + -0.7) equal to 1.044030651 and -1.044030651.

Subproblem 1

x + -0.7 = 1.044030651 Simplifying x + -0.7 = 1.044030651 Reorder the terms: -0.7 + x = 1.044030651 Solving -0.7 + x = 1.044030651 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + x = 1.044030651 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + x = 1.044030651 + 0.7 x = 1.044030651 + 0.7 Combine like terms: 1.044030651 + 0.7 = 1.744030651 x = 1.744030651 Simplifying x = 1.744030651

Subproblem 2

x + -0.7 = -1.044030651 Simplifying x + -0.7 = -1.044030651 Reorder the terms: -0.7 + x = -1.044030651 Solving -0.7 + x = -1.044030651 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + x = -1.044030651 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + x = -1.044030651 + 0.7 x = -1.044030651 + 0.7 Combine like terms: -1.044030651 + 0.7 = -0.344030651 x = -0.344030651 Simplifying x = -0.344030651

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.744030651, -0.344030651}

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