(-3x^2+4x-10)+(x^2-9x+15)=0

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Solution for (-3x^2+4x-10)+(x^2-9x+15)=0 equation:


Simplifying
(-3x2 + 4x + -10) + (x2 + -9x + 15) = 0

Reorder the terms:
(-10 + 4x + -3x2) + (x2 + -9x + 15) = 0

Remove parenthesis around (-10 + 4x + -3x2)
-10 + 4x + -3x2 + (x2 + -9x + 15) = 0

Reorder the terms:
-10 + 4x + -3x2 + (15 + -9x + x2) = 0

Remove parenthesis around (15 + -9x + x2)
-10 + 4x + -3x2 + 15 + -9x + x2 = 0

Reorder the terms:
-10 + 15 + 4x + -9x + -3x2 + x2 = 0

Combine like terms: -10 + 15 = 5
5 + 4x + -9x + -3x2 + x2 = 0

Combine like terms: 4x + -9x = -5x
5 + -5x + -3x2 + x2 = 0

Combine like terms: -3x2 + x2 = -2x2
5 + -5x + -2x2 = 0

Solving
5 + -5x + -2x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-2.5 + 2.5x + x2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + 2.5x + 2.5 + x2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + 2.5x + x2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + 2.5x + x2 = 0 + 2.5
2.5x + x2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
2.5x + x2 = 2.5

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 = 2.5 + 1.5625

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

Combine like terms: 2.5 + 1.5625 = 4.0625
1.5625 + 2.5x + x2 = 4.0625

Factor a perfect square on the left side:
(x + 1.25)(x + 1.25) = 4.0625

Calculate the square root of the right side: 2.015564437

Break this problem into two subproblems by setting 
(x + 1.25) equal to 2.015564437 and -2.015564437.

Subproblem 1

x + 1.25 = 2.015564437 Simplifying x + 1.25 = 2.015564437 Reorder the terms: 1.25 + x = 2.015564437 Solving 1.25 + x = 2.015564437 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 = 2.015564437 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = 2.015564437 + -1.25 x = 2.015564437 + -1.25 Combine like terms: 2.015564437 + -1.25 = 0.765564437 x = 0.765564437 Simplifying x = 0.765564437

Subproblem 2

x + 1.25 = -2.015564437 Simplifying x + 1.25 = -2.015564437 Reorder the terms: 1.25 + x = -2.015564437 Solving 1.25 + x = -2.015564437 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 = -2.015564437 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = -2.015564437 + -1.25 x = -2.015564437 + -1.25 Combine like terms: -2.015564437 + -1.25 = -3.265564437 x = -3.265564437 Simplifying x = -3.265564437

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.765564437, -3.265564437}

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