(3x-1)(3-x)+4x^2-19=0

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Solution for (3x-1)(3-x)+4x^2-19=0 equation:


Simplifying
(3x + -1)(3 + -1x) + 4x2 + -19 = 0

Reorder the terms:
(-1 + 3x)(3 + -1x) + 4x2 + -19 = 0

Multiply (-1 + 3x) * (3 + -1x)
(-1(3 + -1x) + 3x * (3 + -1x)) + 4x2 + -19 = 0
((3 * -1 + -1x * -1) + 3x * (3 + -1x)) + 4x2 + -19 = 0
((-3 + 1x) + 3x * (3 + -1x)) + 4x2 + -19 = 0
(-3 + 1x + (3 * 3x + -1x * 3x)) + 4x2 + -19 = 0
(-3 + 1x + (9x + -3x2)) + 4x2 + -19 = 0

Combine like terms: 1x + 9x = 10x
(-3 + 10x + -3x2) + 4x2 + -19 = 0

Reorder the terms:
-3 + -19 + 10x + -3x2 + 4x2 = 0

Combine like terms: -3 + -19 = -22
-22 + 10x + -3x2 + 4x2 = 0

Combine like terms: -3x2 + 4x2 = 1x2
-22 + 10x + 1x2 = 0

Solving
-22 + 10x + 1x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '22' to each side of the equation.
-22 + 10x + 22 + x2 = 0 + 22

Reorder the terms:
-22 + 22 + 10x + x2 = 0 + 22

Combine like terms: -22 + 22 = 0
0 + 10x + x2 = 0 + 22
10x + x2 = 0 + 22

Combine like terms: 0 + 22 = 22
10x + x2 = 22

The x term is 10x.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10x + 25 + x2 = 22 + 25

Reorder the terms:
25 + 10x + x2 = 22 + 25

Combine like terms: 22 + 25 = 47
25 + 10x + x2 = 47

Factor a perfect square on the left side:
(x + 5)(x + 5) = 47

Calculate the square root of the right side: 6.8556546

Break this problem into two subproblems by setting 
(x + 5) equal to 6.8556546 and -6.8556546.

Subproblem 1

x + 5 = 6.8556546 Simplifying x + 5 = 6.8556546 Reorder the terms: 5 + x = 6.8556546 Solving 5 + x = 6.8556546 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + x = 6.8556546 + -5 Combine like terms: 5 + -5 = 0 0 + x = 6.8556546 + -5 x = 6.8556546 + -5 Combine like terms: 6.8556546 + -5 = 1.8556546 x = 1.8556546 Simplifying x = 1.8556546

Subproblem 2

x + 5 = -6.8556546 Simplifying x + 5 = -6.8556546 Reorder the terms: 5 + x = -6.8556546 Solving 5 + x = -6.8556546 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + x = -6.8556546 + -5 Combine like terms: 5 + -5 = 0 0 + x = -6.8556546 + -5 x = -6.8556546 + -5 Combine like terms: -6.8556546 + -5 = -11.8556546 x = -11.8556546 Simplifying x = -11.8556546

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.8556546, -11.8556546}

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