2x^2-7.5x+x^2=8.5

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Solution for 2x^2-7.5x+x^2=8.5 equation:


Simplifying
2x2 + -7.5x + x2 = 8.5

Reorder the terms:
-7.5x + 2x2 + x2 = 8.5

Combine like terms: 2x2 + x2 = 3x2
-7.5x + 3x2 = 8.5

Solving
-7.5x + 3x2 = 8.5

Solving for variable 'x'.

Reorder the terms:
-8.5 + -7.5x + 3x2 = 8.5 + -8.5

Combine like terms: 8.5 + -8.5 = 0.0
-8.5 + -7.5x + 3x2 = 0.0

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

Divide each side by '3'.
-2.833333333 + -2.5x + x2 = 0.0

Move the constant term to the right:

Add '2.833333333' to each side of the equation.
-2.833333333 + -2.5x + 2.833333333 + x2 = 0.0 + 2.833333333

Reorder the terms:
-2.833333333 + 2.833333333 + -2.5x + x2 = 0.0 + 2.833333333

Combine like terms: -2.833333333 + 2.833333333 = 0.000000000
0.000000000 + -2.5x + x2 = 0.0 + 2.833333333
-2.5x + x2 = 0.0 + 2.833333333

Combine like terms: 0.0 + 2.833333333 = 2.833333333
-2.5x + x2 = 2.833333333

The x term is -2.5x.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x + 1.5625 + x2 = 2.833333333 + 1.5625

Reorder the terms:
1.5625 + -2.5x + x2 = 2.833333333 + 1.5625

Combine like terms: 2.833333333 + 1.5625 = 4.395833333
1.5625 + -2.5x + x2 = 4.395833333

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 4.395833333

Calculate the square root of the right side: 2.096624271

Break this problem into two subproblems by setting 
(x + -1.25) equal to 2.096624271 and -2.096624271.

Subproblem 1

x + -1.25 = 2.096624271 Simplifying x + -1.25 = 2.096624271 Reorder the terms: -1.25 + x = 2.096624271 Solving -1.25 + x = 2.096624271 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 2.096624271 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 2.096624271 + 1.25 x = 2.096624271 + 1.25 Combine like terms: 2.096624271 + 1.25 = 3.346624271 x = 3.346624271 Simplifying x = 3.346624271

Subproblem 2

x + -1.25 = -2.096624271 Simplifying x + -1.25 = -2.096624271 Reorder the terms: -1.25 + x = -2.096624271 Solving -1.25 + x = -2.096624271 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -2.096624271 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -2.096624271 + 1.25 x = -2.096624271 + 1.25 Combine like terms: -2.096624271 + 1.25 = -0.846624271 x = -0.846624271 Simplifying x = -0.846624271

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.346624271, -0.846624271}

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