(x-10)(2x-3)+20=5x^2

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Solution for (x-10)(2x-3)+20=5x^2 equation:


Simplifying
(x + -10)(2x + -3) + 20 = 5x2

Reorder the terms:
(-10 + x)(2x + -3) + 20 = 5x2

Reorder the terms:
(-10 + x)(-3 + 2x) + 20 = 5x2

Multiply (-10 + x) * (-3 + 2x)
(-10(-3 + 2x) + x(-3 + 2x)) + 20 = 5x2
((-3 * -10 + 2x * -10) + x(-3 + 2x)) + 20 = 5x2
((30 + -20x) + x(-3 + 2x)) + 20 = 5x2
(30 + -20x + (-3 * x + 2x * x)) + 20 = 5x2
(30 + -20x + (-3x + 2x2)) + 20 = 5x2

Combine like terms: -20x + -3x = -23x
(30 + -23x + 2x2) + 20 = 5x2

Reorder the terms:
30 + 20 + -23x + 2x2 = 5x2

Combine like terms: 30 + 20 = 50
50 + -23x + 2x2 = 5x2

Solving
50 + -23x + 2x2 = 5x2

Solving for variable 'x'.

Combine like terms: 2x2 + -5x2 = -3x2
50 + -23x + -3x2 = 5x2 + -5x2

Combine like terms: 5x2 + -5x2 = 0
50 + -23x + -3x2 = 0

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

Divide each side by '-3'.
-16.66666667 + 7.666666667x + x2 = 0

Move the constant term to the right:

Add '16.66666667' to each side of the equation.
-16.66666667 + 7.666666667x + 16.66666667 + x2 = 0 + 16.66666667

Reorder the terms:
-16.66666667 + 16.66666667 + 7.666666667x + x2 = 0 + 16.66666667

Combine like terms: -16.66666667 + 16.66666667 = 0.00000000
0.00000000 + 7.666666667x + x2 = 0 + 16.66666667
7.666666667x + x2 = 0 + 16.66666667

Combine like terms: 0 + 16.66666667 = 16.66666667
7.666666667x + x2 = 16.66666667

The x term is 7.666666667x.  Take half its coefficient (3.833333334).
Square it (14.69444445) and add it to both sides.

Add '14.69444445' to each side of the equation.
7.666666667x + 14.69444445 + x2 = 16.66666667 + 14.69444445

Reorder the terms:
14.69444445 + 7.666666667x + x2 = 16.66666667 + 14.69444445

Combine like terms: 16.66666667 + 14.69444445 = 31.36111112
14.69444445 + 7.666666667x + x2 = 31.36111112

Factor a perfect square on the left side:
(x + 3.833333334)(x + 3.833333334) = 31.36111112

Calculate the square root of the right side: 5.600099206

Break this problem into two subproblems by setting 
(x + 3.833333334) equal to 5.600099206 and -5.600099206.

Subproblem 1

x + 3.833333334 = 5.600099206 Simplifying x + 3.833333334 = 5.600099206 Reorder the terms: 3.833333334 + x = 5.600099206 Solving 3.833333334 + x = 5.600099206 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.833333334' to each side of the equation. 3.833333334 + -3.833333334 + x = 5.600099206 + -3.833333334 Combine like terms: 3.833333334 + -3.833333334 = 0.000000000 0.000000000 + x = 5.600099206 + -3.833333334 x = 5.600099206 + -3.833333334 Combine like terms: 5.600099206 + -3.833333334 = 1.766765872 x = 1.766765872 Simplifying x = 1.766765872

Subproblem 2

x + 3.833333334 = -5.600099206 Simplifying x + 3.833333334 = -5.600099206 Reorder the terms: 3.833333334 + x = -5.600099206 Solving 3.833333334 + x = -5.600099206 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.833333334' to each side of the equation. 3.833333334 + -3.833333334 + x = -5.600099206 + -3.833333334 Combine like terms: 3.833333334 + -3.833333334 = 0.000000000 0.000000000 + x = -5.600099206 + -3.833333334 x = -5.600099206 + -3.833333334 Combine like terms: -5.600099206 + -3.833333334 = -9.43343254 x = -9.43343254 Simplifying x = -9.43343254

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.766765872, -9.43343254}

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