(3x-7)(x-4)=9

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Solution for (3x-7)(x-4)=9 equation:


Simplifying
(3x + -7)(x + -4) = 9

Reorder the terms:
(-7 + 3x)(x + -4) = 9

Reorder the terms:
(-7 + 3x)(-4 + x) = 9

Multiply (-7 + 3x) * (-4 + x)
(-7(-4 + x) + 3x * (-4 + x)) = 9
((-4 * -7 + x * -7) + 3x * (-4 + x)) = 9
((28 + -7x) + 3x * (-4 + x)) = 9
(28 + -7x + (-4 * 3x + x * 3x)) = 9
(28 + -7x + (-12x + 3x2)) = 9

Combine like terms: -7x + -12x = -19x
(28 + -19x + 3x2) = 9

Solving
28 + -19x + 3x2 = 9

Solving for variable 'x'.

Reorder the terms:
28 + -9 + -19x + 3x2 = 9 + -9

Combine like terms: 28 + -9 = 19
19 + -19x + 3x2 = 9 + -9

Combine like terms: 9 + -9 = 0
19 + -19x + 3x2 = 0

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

Divide each side by '3'.
6.333333333 + -6.333333333x + x2 = 0

Move the constant term to the right:

Add '-6.333333333' to each side of the equation.
6.333333333 + -6.333333333x + -6.333333333 + x2 = 0 + -6.333333333

Reorder the terms:
6.333333333 + -6.333333333 + -6.333333333x + x2 = 0 + -6.333333333

Combine like terms: 6.333333333 + -6.333333333 = 0.000000000
0.000000000 + -6.333333333x + x2 = 0 + -6.333333333
-6.333333333x + x2 = 0 + -6.333333333

Combine like terms: 0 + -6.333333333 = -6.333333333
-6.333333333x + x2 = -6.333333333

The x term is -6.333333333x.  Take half its coefficient (-3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
-6.333333333x + 10.02777778 + x2 = -6.333333333 + 10.02777778

Reorder the terms:
10.02777778 + -6.333333333x + x2 = -6.333333333 + 10.02777778

Combine like terms: -6.333333333 + 10.02777778 = 3.694444447
10.02777778 + -6.333333333x + x2 = 3.694444447

Factor a perfect square on the left side:
(x + -3.166666667)(x + -3.166666667) = 3.694444447

Calculate the square root of the right side: 1.922093766

Break this problem into two subproblems by setting 
(x + -3.166666667) equal to 1.922093766 and -1.922093766.

Subproblem 1

x + -3.166666667 = 1.922093766 Simplifying x + -3.166666667 = 1.922093766 Reorder the terms: -3.166666667 + x = 1.922093766 Solving -3.166666667 + x = 1.922093766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.166666667' to each side of the equation. -3.166666667 + 3.166666667 + x = 1.922093766 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + x = 1.922093766 + 3.166666667 x = 1.922093766 + 3.166666667 Combine like terms: 1.922093766 + 3.166666667 = 5.088760433 x = 5.088760433 Simplifying x = 5.088760433

Subproblem 2

x + -3.166666667 = -1.922093766 Simplifying x + -3.166666667 = -1.922093766 Reorder the terms: -3.166666667 + x = -1.922093766 Solving -3.166666667 + x = -1.922093766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.166666667' to each side of the equation. -3.166666667 + 3.166666667 + x = -1.922093766 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + x = -1.922093766 + 3.166666667 x = -1.922093766 + 3.166666667 Combine like terms: -1.922093766 + 3.166666667 = 1.244572901 x = 1.244572901 Simplifying x = 1.244572901

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.088760433, 1.244572901}

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