0=x^2-3x-42

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Solution for 0=x^2-3x-42 equation:


Simplifying
0 = x2 + -3x + -42

Reorder the terms:
0 = -42 + -3x + x2

Solving
0 = -42 + -3x + x2

Solving for variable 'x'.

Combine like terms: 0 + 42 = 42
42 + 3x + -1x2 = -42 + -3x + x2 + 42 + 3x + -1x2

Reorder the terms:
42 + 3x + -1x2 = -42 + 42 + -3x + 3x + x2 + -1x2

Combine like terms: -42 + 42 = 0
42 + 3x + -1x2 = 0 + -3x + 3x + x2 + -1x2
42 + 3x + -1x2 = -3x + 3x + x2 + -1x2

Combine like terms: -3x + 3x = 0
42 + 3x + -1x2 = 0 + x2 + -1x2
42 + 3x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
42 + 3x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-42 + -3x + x2 = 0

Move the constant term to the right:

Add '42' to each side of the equation.
-42 + -3x + 42 + x2 = 0 + 42

Reorder the terms:
-42 + 42 + -3x + x2 = 0 + 42

Combine like terms: -42 + 42 = 0
0 + -3x + x2 = 0 + 42
-3x + x2 = 0 + 42

Combine like terms: 0 + 42 = 42
-3x + x2 = 42

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 42 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 42 + 2.25

Combine like terms: 42 + 2.25 = 44.25
2.25 + -3x + x2 = 44.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 44.25

Calculate the square root of the right side: 6.652067348

Break this problem into two subproblems by setting 
(x + -1.5) equal to 6.652067348 and -6.652067348.

Subproblem 1

x + -1.5 = 6.652067348 Simplifying x + -1.5 = 6.652067348 Reorder the terms: -1.5 + x = 6.652067348 Solving -1.5 + x = 6.652067348 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 6.652067348 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 6.652067348 + 1.5 x = 6.652067348 + 1.5 Combine like terms: 6.652067348 + 1.5 = 8.152067348 x = 8.152067348 Simplifying x = 8.152067348

Subproblem 2

x + -1.5 = -6.652067348 Simplifying x + -1.5 = -6.652067348 Reorder the terms: -1.5 + x = -6.652067348 Solving -1.5 + x = -6.652067348 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -6.652067348 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -6.652067348 + 1.5 x = -6.652067348 + 1.5 Combine like terms: -6.652067348 + 1.5 = -5.152067348 x = -5.152067348 Simplifying x = -5.152067348

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.152067348, -5.152067348}

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