10(x^2-x-6)=1

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Solution for 10(x^2-x-6)=1 equation:


Simplifying
10(x2 + -1x + -6) = 1

Reorder the terms:
10(-6 + -1x + x2) = 1
(-6 * 10 + -1x * 10 + x2 * 10) = 1
(-60 + -10x + 10x2) = 1

Solving
-60 + -10x + 10x2 = 1

Solving for variable 'x'.

Reorder the terms:
-60 + -1 + -10x + 10x2 = 1 + -1

Combine like terms: -60 + -1 = -61
-61 + -10x + 10x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-61 + -10x + 10x2 = 0

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

Divide each side by '10'.
-6.1 + -1x + x2 = 0

Move the constant term to the right:

Add '6.1' to each side of the equation.
-6.1 + -1x + 6.1 + x2 = 0 + 6.1

Reorder the terms:
-6.1 + 6.1 + -1x + x2 = 0 + 6.1

Combine like terms: -6.1 + 6.1 = 0.0
0.0 + -1x + x2 = 0 + 6.1
-1x + x2 = 0 + 6.1

Combine like terms: 0 + 6.1 = 6.1
-1x + x2 = 6.1

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 6.1 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 6.1 + 0.25

Combine like terms: 6.1 + 0.25 = 6.35
0.25 + -1x + x2 = 6.35

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 6.35

Calculate the square root of the right side: 2.519920634

Break this problem into two subproblems by setting 
(x + -0.5) equal to 2.519920634 and -2.519920634.

Subproblem 1

x + -0.5 = 2.519920634 Simplifying x + -0.5 = 2.519920634 Reorder the terms: -0.5 + x = 2.519920634 Solving -0.5 + x = 2.519920634 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 2.519920634 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 2.519920634 + 0.5 x = 2.519920634 + 0.5 Combine like terms: 2.519920634 + 0.5 = 3.019920634 x = 3.019920634 Simplifying x = 3.019920634

Subproblem 2

x + -0.5 = -2.519920634 Simplifying x + -0.5 = -2.519920634 Reorder the terms: -0.5 + x = -2.519920634 Solving -0.5 + x = -2.519920634 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -2.519920634 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -2.519920634 + 0.5 x = -2.519920634 + 0.5 Combine like terms: -2.519920634 + 0.5 = -2.019920634 x = -2.019920634 Simplifying x = -2.019920634

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.019920634, -2.019920634}

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