5x^2-x-15=3x^2+1

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Solution for 5x^2-x-15=3x^2+1 equation:


Simplifying
5x2 + -1x + -15 = 3x2 + 1

Reorder the terms:
-15 + -1x + 5x2 = 3x2 + 1

Reorder the terms:
-15 + -1x + 5x2 = 1 + 3x2

Solving
-15 + -1x + 5x2 = 1 + 3x2

Solving for variable 'x'.

Reorder the terms:
-15 + -1 + -1x + 5x2 + -3x2 = 1 + 3x2 + -1 + -3x2

Combine like terms: -15 + -1 = -16
-16 + -1x + 5x2 + -3x2 = 1 + 3x2 + -1 + -3x2

Combine like terms: 5x2 + -3x2 = 2x2
-16 + -1x + 2x2 = 1 + 3x2 + -1 + -3x2

Reorder the terms:
-16 + -1x + 2x2 = 1 + -1 + 3x2 + -3x2

Combine like terms: 1 + -1 = 0
-16 + -1x + 2x2 = 0 + 3x2 + -3x2
-16 + -1x + 2x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
-16 + -1x + 2x2 = 0

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

Divide each side by '2'.
-8 + -0.5x + x2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -0.5x + 8 + x2 = 0 + 8

Reorder the terms:
-8 + 8 + -0.5x + x2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + -0.5x + x2 = 0 + 8
-0.5x + x2 = 0 + 8

Combine like terms: 0 + 8 = 8
-0.5x + x2 = 8

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 8 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 8 + 0.0625

Combine like terms: 8 + 0.0625 = 8.0625
0.0625 + -0.5x + x2 = 8.0625

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 8.0625

Calculate the square root of the right side: 2.839454173

Break this problem into two subproblems by setting 
(x + -0.25) equal to 2.839454173 and -2.839454173.

Subproblem 1

x + -0.25 = 2.839454173 Simplifying x + -0.25 = 2.839454173 Reorder the terms: -0.25 + x = 2.839454173 Solving -0.25 + x = 2.839454173 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 2.839454173 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 2.839454173 + 0.25 x = 2.839454173 + 0.25 Combine like terms: 2.839454173 + 0.25 = 3.089454173 x = 3.089454173 Simplifying x = 3.089454173

Subproblem 2

x + -0.25 = -2.839454173 Simplifying x + -0.25 = -2.839454173 Reorder the terms: -0.25 + x = -2.839454173 Solving -0.25 + x = -2.839454173 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -2.839454173 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -2.839454173 + 0.25 x = -2.839454173 + 0.25 Combine like terms: -2.839454173 + 0.25 = -2.589454173 x = -2.589454173 Simplifying x = -2.589454173

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.089454173, -2.589454173}

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