(7x+3)(4x+1)=65

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Solution for (7x+3)(4x+1)=65 equation:


Simplifying
(7x + 3)(4x + 1) = 65

Reorder the terms:
(3 + 7x)(4x + 1) = 65

Reorder the terms:
(3 + 7x)(1 + 4x) = 65

Multiply (3 + 7x) * (1 + 4x)
(3(1 + 4x) + 7x * (1 + 4x)) = 65
((1 * 3 + 4x * 3) + 7x * (1 + 4x)) = 65
((3 + 12x) + 7x * (1 + 4x)) = 65
(3 + 12x + (1 * 7x + 4x * 7x)) = 65
(3 + 12x + (7x + 28x2)) = 65

Combine like terms: 12x + 7x = 19x
(3 + 19x + 28x2) = 65

Solving
3 + 19x + 28x2 = 65

Solving for variable 'x'.

Reorder the terms:
3 + -65 + 19x + 28x2 = 65 + -65

Combine like terms: 3 + -65 = -62
-62 + 19x + 28x2 = 65 + -65

Combine like terms: 65 + -65 = 0
-62 + 19x + 28x2 = 0

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

Divide each side by '28'.
-2.214285714 + 0.6785714286x + x2 = 0

Move the constant term to the right:

Add '2.214285714' to each side of the equation.
-2.214285714 + 0.6785714286x + 2.214285714 + x2 = 0 + 2.214285714

Reorder the terms:
-2.214285714 + 2.214285714 + 0.6785714286x + x2 = 0 + 2.214285714

Combine like terms: -2.214285714 + 2.214285714 = 0.000000000
0.000000000 + 0.6785714286x + x2 = 0 + 2.214285714
0.6785714286x + x2 = 0 + 2.214285714

Combine like terms: 0 + 2.214285714 = 2.214285714
0.6785714286x + x2 = 2.214285714

The x term is 0.6785714286x.  Take half its coefficient (0.3392857143).
Square it (0.1151147959) and add it to both sides.

Add '0.1151147959' to each side of the equation.
0.6785714286x + 0.1151147959 + x2 = 2.214285714 + 0.1151147959

Reorder the terms:
0.1151147959 + 0.6785714286x + x2 = 2.214285714 + 0.1151147959

Combine like terms: 2.214285714 + 0.1151147959 = 2.3294005099
0.1151147959 + 0.6785714286x + x2 = 2.3294005099

Factor a perfect square on the left side:
(x + 0.3392857143)(x + 0.3392857143) = 2.3294005099

Calculate the square root of the right side: 1.52623737

Break this problem into two subproblems by setting 
(x + 0.3392857143) equal to 1.52623737 and -1.52623737.

Subproblem 1

x + 0.3392857143 = 1.52623737 Simplifying x + 0.3392857143 = 1.52623737 Reorder the terms: 0.3392857143 + x = 1.52623737 Solving 0.3392857143 + x = 1.52623737 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3392857143' to each side of the equation. 0.3392857143 + -0.3392857143 + x = 1.52623737 + -0.3392857143 Combine like terms: 0.3392857143 + -0.3392857143 = 0.0000000000 0.0000000000 + x = 1.52623737 + -0.3392857143 x = 1.52623737 + -0.3392857143 Combine like terms: 1.52623737 + -0.3392857143 = 1.1869516557 x = 1.1869516557 Simplifying x = 1.1869516557

Subproblem 2

x + 0.3392857143 = -1.52623737 Simplifying x + 0.3392857143 = -1.52623737 Reorder the terms: 0.3392857143 + x = -1.52623737 Solving 0.3392857143 + x = -1.52623737 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3392857143' to each side of the equation. 0.3392857143 + -0.3392857143 + x = -1.52623737 + -0.3392857143 Combine like terms: 0.3392857143 + -0.3392857143 = 0.0000000000 0.0000000000 + x = -1.52623737 + -0.3392857143 x = -1.52623737 + -0.3392857143 Combine like terms: -1.52623737 + -0.3392857143 = -1.8655230843 x = -1.8655230843 Simplifying x = -1.8655230843

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.1869516557, -1.8655230843}

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