7x^2=4x+7

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Solution for 7x^2=4x+7 equation:


Simplifying
7x2 = 4x + 7

Reorder the terms:
7x2 = 7 + 4x

Solving
7x2 = 7 + 4x

Solving for variable 'x'.

Reorder the terms:
-7 + -4x + 7x2 = 7 + 4x + -7 + -4x

Reorder the terms:
-7 + -4x + 7x2 = 7 + -7 + 4x + -4x

Combine like terms: 7 + -7 = 0
-7 + -4x + 7x2 = 0 + 4x + -4x
-7 + -4x + 7x2 = 4x + -4x

Combine like terms: 4x + -4x = 0
-7 + -4x + 7x2 = 0

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

Divide each side by '7'.
-1 + -0.5714285714x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.5714285714x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.5714285714x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.5714285714x + x2 = 0 + 1
-0.5714285714x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.5714285714x + x2 = 1

The x term is -0.5714285714x.  Take half its coefficient (-0.2857142857).
Square it (0.08163265305) and add it to both sides.

Add '0.08163265305' to each side of the equation.
-0.5714285714x + 0.08163265305 + x2 = 1 + 0.08163265305

Reorder the terms:
0.08163265305 + -0.5714285714x + x2 = 1 + 0.08163265305

Combine like terms: 1 + 0.08163265305 = 1.08163265305
0.08163265305 + -0.5714285714x + x2 = 1.08163265305

Factor a perfect square on the left side:
(x + -0.2857142857)(x + -0.2857142857) = 1.08163265305

Calculate the square root of the right side: 1.040015698

Break this problem into two subproblems by setting 
(x + -0.2857142857) equal to 1.040015698 and -1.040015698.

Subproblem 1

x + -0.2857142857 = 1.040015698 Simplifying x + -0.2857142857 = 1.040015698 Reorder the terms: -0.2857142857 + x = 1.040015698 Solving -0.2857142857 + x = 1.040015698 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + x = 1.040015698 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + x = 1.040015698 + 0.2857142857 x = 1.040015698 + 0.2857142857 Combine like terms: 1.040015698 + 0.2857142857 = 1.3257299837 x = 1.3257299837 Simplifying x = 1.3257299837

Subproblem 2

x + -0.2857142857 = -1.040015698 Simplifying x + -0.2857142857 = -1.040015698 Reorder the terms: -0.2857142857 + x = -1.040015698 Solving -0.2857142857 + x = -1.040015698 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + x = -1.040015698 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + x = -1.040015698 + 0.2857142857 x = -1.040015698 + 0.2857142857 Combine like terms: -1.040015698 + 0.2857142857 = -0.7543014123 x = -0.7543014123 Simplifying x = -0.7543014123

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.3257299837, -0.7543014123}

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