3.5x+0.75(x-1)x-24=0

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Solution for 3.5x+0.75(x-1)x-24=0 equation:


Simplifying
3.5x + 0.75(x + -1) * x + -24 = 0

Reorder the terms:
3.5x + 0.75(-1 + x) * x + -24 = 0

Reorder the terms for easier multiplication:
3.5x + 0.75x(-1 + x) + -24 = 0
3.5x + (-1 * 0.75x + x * 0.75x) + -24 = 0
3.5x + (-0.75x + 0.75x2) + -24 = 0

Reorder the terms:
-24 + 3.5x + -0.75x + 0.75x2 = 0

Combine like terms: 3.5x + -0.75x = 2.75x
-24 + 2.75x + 0.75x2 = 0

Solving
-24 + 2.75x + 0.75x2 = 0

Solving for variable 'x'.

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

Divide each side by '0.75'.
-32 + 3.666666667x + x2 = 0

Move the constant term to the right:

Add '32' to each side of the equation.
-32 + 3.666666667x + 32 + x2 = 0 + 32

Reorder the terms:
-32 + 32 + 3.666666667x + x2 = 0 + 32

Combine like terms: -32 + 32 = 0
0 + 3.666666667x + x2 = 0 + 32
3.666666667x + x2 = 0 + 32

Combine like terms: 0 + 32 = 32
3.666666667x + x2 = 32

The x term is 3.666666667x.  Take half its coefficient (1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
3.666666667x + 3.361111114 + x2 = 32 + 3.361111114

Reorder the terms:
3.361111114 + 3.666666667x + x2 = 32 + 3.361111114

Combine like terms: 32 + 3.361111114 = 35.361111114
3.361111114 + 3.666666667x + x2 = 35.361111114

Factor a perfect square on the left side:
(x + 1.833333334)(x + 1.833333334) = 35.361111114

Calculate the square root of the right side: 5.946520925

Break this problem into two subproblems by setting 
(x + 1.833333334) equal to 5.946520925 and -5.946520925.

Subproblem 1

x + 1.833333334 = 5.946520925 Simplifying x + 1.833333334 = 5.946520925 Reorder the terms: 1.833333334 + x = 5.946520925 Solving 1.833333334 + x = 5.946520925 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + x = 5.946520925 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = 5.946520925 + -1.833333334 x = 5.946520925 + -1.833333334 Combine like terms: 5.946520925 + -1.833333334 = 4.113187591 x = 4.113187591 Simplifying x = 4.113187591

Subproblem 2

x + 1.833333334 = -5.946520925 Simplifying x + 1.833333334 = -5.946520925 Reorder the terms: 1.833333334 + x = -5.946520925 Solving 1.833333334 + x = -5.946520925 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + x = -5.946520925 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = -5.946520925 + -1.833333334 x = -5.946520925 + -1.833333334 Combine like terms: -5.946520925 + -1.833333334 = -7.779854259 x = -7.779854259 Simplifying x = -7.779854259

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.113187591, -7.779854259}

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