(13x-6)(5x+14)=54

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Solution for (13x-6)(5x+14)=54 equation:


Simplifying
(13x + -6)(5x + 14) = 54

Reorder the terms:
(-6 + 13x)(5x + 14) = 54

Reorder the terms:
(-6 + 13x)(14 + 5x) = 54

Multiply (-6 + 13x) * (14 + 5x)
(-6(14 + 5x) + 13x * (14 + 5x)) = 54
((14 * -6 + 5x * -6) + 13x * (14 + 5x)) = 54
((-84 + -30x) + 13x * (14 + 5x)) = 54
(-84 + -30x + (14 * 13x + 5x * 13x)) = 54
(-84 + -30x + (182x + 65x2)) = 54

Combine like terms: -30x + 182x = 152x
(-84 + 152x + 65x2) = 54

Solving
-84 + 152x + 65x2 = 54

Solving for variable 'x'.

Reorder the terms:
-84 + -54 + 152x + 65x2 = 54 + -54

Combine like terms: -84 + -54 = -138
-138 + 152x + 65x2 = 54 + -54

Combine like terms: 54 + -54 = 0
-138 + 152x + 65x2 = 0

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

Divide each side by '65'.
-2.123076923 + 2.338461538x + x2 = 0

Move the constant term to the right:

Add '2.123076923' to each side of the equation.
-2.123076923 + 2.338461538x + 2.123076923 + x2 = 0 + 2.123076923

Reorder the terms:
-2.123076923 + 2.123076923 + 2.338461538x + x2 = 0 + 2.123076923

Combine like terms: -2.123076923 + 2.123076923 = 0.000000000
0.000000000 + 2.338461538x + x2 = 0 + 2.123076923
2.338461538x + x2 = 0 + 2.123076923

Combine like terms: 0 + 2.123076923 = 2.123076923
2.338461538x + x2 = 2.123076923

The x term is 2.338461538x.  Take half its coefficient (1.169230769).
Square it (1.367100591) and add it to both sides.

Add '1.367100591' to each side of the equation.
2.338461538x + 1.367100591 + x2 = 2.123076923 + 1.367100591

Reorder the terms:
1.367100591 + 2.338461538x + x2 = 2.123076923 + 1.367100591

Combine like terms: 2.123076923 + 1.367100591 = 3.490177514
1.367100591 + 2.338461538x + x2 = 3.490177514

Factor a perfect square on the left side:
(x + 1.169230769)(x + 1.169230769) = 3.490177514

Calculate the square root of the right side: 1.868201679

Break this problem into two subproblems by setting 
(x + 1.169230769) equal to 1.868201679 and -1.868201679.

Subproblem 1

x + 1.169230769 = 1.868201679 Simplifying x + 1.169230769 = 1.868201679 Reorder the terms: 1.169230769 + x = 1.868201679 Solving 1.169230769 + x = 1.868201679 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.169230769' to each side of the equation. 1.169230769 + -1.169230769 + x = 1.868201679 + -1.169230769 Combine like terms: 1.169230769 + -1.169230769 = 0.000000000 0.000000000 + x = 1.868201679 + -1.169230769 x = 1.868201679 + -1.169230769 Combine like terms: 1.868201679 + -1.169230769 = 0.69897091 x = 0.69897091 Simplifying x = 0.69897091

Subproblem 2

x + 1.169230769 = -1.868201679 Simplifying x + 1.169230769 = -1.868201679 Reorder the terms: 1.169230769 + x = -1.868201679 Solving 1.169230769 + x = -1.868201679 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.169230769' to each side of the equation. 1.169230769 + -1.169230769 + x = -1.868201679 + -1.169230769 Combine like terms: 1.169230769 + -1.169230769 = 0.000000000 0.000000000 + x = -1.868201679 + -1.169230769 x = -1.868201679 + -1.169230769 Combine like terms: -1.868201679 + -1.169230769 = -3.037432448 x = -3.037432448 Simplifying x = -3.037432448

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.69897091, -3.037432448}

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