4x^2-28x-75=0

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Solution for 4x^2-28x-75=0 equation:


Simplifying
4x2 + -28x + -75 = 0

Reorder the terms:
-75 + -28x + 4x2 = 0

Solving
-75 + -28x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
-18.75 + -7x + x2 = 0

Move the constant term to the right:

Add '18.75' to each side of the equation.
-18.75 + -7x + 18.75 + x2 = 0 + 18.75

Reorder the terms:
-18.75 + 18.75 + -7x + x2 = 0 + 18.75

Combine like terms: -18.75 + 18.75 = 0.00
0.00 + -7x + x2 = 0 + 18.75
-7x + x2 = 0 + 18.75

Combine like terms: 0 + 18.75 = 18.75
-7x + x2 = 18.75

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = 18.75 + 12.25

Reorder the terms:
12.25 + -7x + x2 = 18.75 + 12.25

Combine like terms: 18.75 + 12.25 = 31
12.25 + -7x + x2 = 31

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 31

Calculate the square root of the right side: 5.567764363

Break this problem into two subproblems by setting 
(x + -3.5) equal to 5.567764363 and -5.567764363.

Subproblem 1

x + -3.5 = 5.567764363 Simplifying x + -3.5 = 5.567764363 Reorder the terms: -3.5 + x = 5.567764363 Solving -3.5 + x = 5.567764363 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = 5.567764363 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 5.567764363 + 3.5 x = 5.567764363 + 3.5 Combine like terms: 5.567764363 + 3.5 = 9.067764363 x = 9.067764363 Simplifying x = 9.067764363

Subproblem 2

x + -3.5 = -5.567764363 Simplifying x + -3.5 = -5.567764363 Reorder the terms: -3.5 + x = -5.567764363 Solving -3.5 + x = -5.567764363 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = -5.567764363 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -5.567764363 + 3.5 x = -5.567764363 + 3.5 Combine like terms: -5.567764363 + 3.5 = -2.067764363 x = -2.067764363 Simplifying x = -2.067764363

Solution

The solution to the problem is based on the solutions from the subproblems. x = {9.067764363, -2.067764363}

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