(4x-5)*(2x-5)=105

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Solution for (4x-5)*(2x-5)=105 equation:


Simplifying
(4x + -5)(2x + -5) = 105

Reorder the terms:
(-5 + 4x)(2x + -5) = 105

Reorder the terms:
(-5 + 4x)(-5 + 2x) = 105

Multiply (-5 + 4x) * (-5 + 2x)
(-5(-5 + 2x) + 4x * (-5 + 2x)) = 105
((-5 * -5 + 2x * -5) + 4x * (-5 + 2x)) = 105
((25 + -10x) + 4x * (-5 + 2x)) = 105
(25 + -10x + (-5 * 4x + 2x * 4x)) = 105
(25 + -10x + (-20x + 8x2)) = 105

Combine like terms: -10x + -20x = -30x
(25 + -30x + 8x2) = 105

Solving
25 + -30x + 8x2 = 105

Solving for variable 'x'.

Reorder the terms:
25 + -105 + -30x + 8x2 = 105 + -105

Combine like terms: 25 + -105 = -80
-80 + -30x + 8x2 = 105 + -105

Combine like terms: 105 + -105 = 0
-80 + -30x + 8x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-40 + -15x + 4x2) = 0

Ignore the factor 2.

Subproblem 1

Set the factor '(-40 + -15x + 4x2)' equal to zero and attempt to solve: Simplifying -40 + -15x + 4x2 = 0 Solving -40 + -15x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -10 + -3.75x + x2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + -3.75x + 10 + x2 = 0 + 10 Reorder the terms: -10 + 10 + -3.75x + x2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + -3.75x + x2 = 0 + 10 -3.75x + x2 = 0 + 10 Combine like terms: 0 + 10 = 10 -3.75x + x2 = 10 The x term is -3.75x. Take half its coefficient (-1.875). Square it (3.515625) and add it to both sides. Add '3.515625' to each side of the equation. -3.75x + 3.515625 + x2 = 10 + 3.515625 Reorder the terms: 3.515625 + -3.75x + x2 = 10 + 3.515625 Combine like terms: 10 + 3.515625 = 13.515625 3.515625 + -3.75x + x2 = 13.515625 Factor a perfect square on the left side: (x + -1.875)(x + -1.875) = 13.515625 Calculate the square root of the right side: 3.676360292 Break this problem into two subproblems by setting (x + -1.875) equal to 3.676360292 and -3.676360292.

Subproblem 1

x + -1.875 = 3.676360292 Simplifying x + -1.875 = 3.676360292 Reorder the terms: -1.875 + x = 3.676360292 Solving -1.875 + x = 3.676360292 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + x = 3.676360292 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + x = 3.676360292 + 1.875 x = 3.676360292 + 1.875 Combine like terms: 3.676360292 + 1.875 = 5.551360292 x = 5.551360292 Simplifying x = 5.551360292

Subproblem 2

x + -1.875 = -3.676360292 Simplifying x + -1.875 = -3.676360292 Reorder the terms: -1.875 + x = -3.676360292 Solving -1.875 + x = -3.676360292 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + x = -3.676360292 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + x = -3.676360292 + 1.875 x = -3.676360292 + 1.875 Combine like terms: -3.676360292 + 1.875 = -1.801360292 x = -1.801360292 Simplifying x = -1.801360292

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.551360292, -1.801360292}

Solution

x = {5.551360292, -1.801360292}

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