7x^2+13x=61

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Solution for 7x^2+13x=61 equation:


Simplifying
7x2 + 13x = 61

Reorder the terms:
13x + 7x2 = 61

Solving
13x + 7x2 = 61

Solving for variable 'x'.

Reorder the terms:
-61 + 13x + 7x2 = 61 + -61

Combine like terms: 61 + -61 = 0
-61 + 13x + 7x2 = 0

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

Divide each side by '7'.
-8.714285714 + 1.857142857x + x2 = 0

Move the constant term to the right:

Add '8.714285714' to each side of the equation.
-8.714285714 + 1.857142857x + 8.714285714 + x2 = 0 + 8.714285714

Reorder the terms:
-8.714285714 + 8.714285714 + 1.857142857x + x2 = 0 + 8.714285714

Combine like terms: -8.714285714 + 8.714285714 = 0.000000000
0.000000000 + 1.857142857x + x2 = 0 + 8.714285714
1.857142857x + x2 = 0 + 8.714285714

Combine like terms: 0 + 8.714285714 = 8.714285714
1.857142857x + x2 = 8.714285714

The x term is 1.857142857x.  Take half its coefficient (0.9285714285).
Square it (0.8622448978) and add it to both sides.

Add '0.8622448978' to each side of the equation.
1.857142857x + 0.8622448978 + x2 = 8.714285714 + 0.8622448978

Reorder the terms:
0.8622448978 + 1.857142857x + x2 = 8.714285714 + 0.8622448978

Combine like terms: 8.714285714 + 0.8622448978 = 9.5765306118
0.8622448978 + 1.857142857x + x2 = 9.5765306118

Factor a perfect square on the left side:
(x + 0.9285714285)(x + 0.9285714285) = 9.5765306118

Calculate the square root of the right side: 3.094597003

Break this problem into two subproblems by setting 
(x + 0.9285714285) equal to 3.094597003 and -3.094597003.

Subproblem 1

x + 0.9285714285 = 3.094597003 Simplifying x + 0.9285714285 = 3.094597003 Reorder the terms: 0.9285714285 + x = 3.094597003 Solving 0.9285714285 + x = 3.094597003 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9285714285' to each side of the equation. 0.9285714285 + -0.9285714285 + x = 3.094597003 + -0.9285714285 Combine like terms: 0.9285714285 + -0.9285714285 = 0.0000000000 0.0000000000 + x = 3.094597003 + -0.9285714285 x = 3.094597003 + -0.9285714285 Combine like terms: 3.094597003 + -0.9285714285 = 2.1660255745 x = 2.1660255745 Simplifying x = 2.1660255745

Subproblem 2

x + 0.9285714285 = -3.094597003 Simplifying x + 0.9285714285 = -3.094597003 Reorder the terms: 0.9285714285 + x = -3.094597003 Solving 0.9285714285 + x = -3.094597003 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9285714285' to each side of the equation. 0.9285714285 + -0.9285714285 + x = -3.094597003 + -0.9285714285 Combine like terms: 0.9285714285 + -0.9285714285 = 0.0000000000 0.0000000000 + x = -3.094597003 + -0.9285714285 x = -3.094597003 + -0.9285714285 Combine like terms: -3.094597003 + -0.9285714285 = -4.0231684315 x = -4.0231684315 Simplifying x = -4.0231684315

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.1660255745, -4.0231684315}

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