0=x^2+3x-7

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Solution for 0=x^2+3x-7 equation:


Simplifying
0 = x2 + 3x + -7

Reorder the terms:
0 = -7 + 3x + x2

Solving
0 = -7 + 3x + x2

Solving for variable 'x'.

Combine like terms: 0 + 7 = 7
7 + -3x + -1x2 = -7 + 3x + x2 + 7 + -3x + -1x2

Reorder the terms:
7 + -3x + -1x2 = -7 + 7 + 3x + -3x + x2 + -1x2

Combine like terms: -7 + 7 = 0
7 + -3x + -1x2 = 0 + 3x + -3x + x2 + -1x2
7 + -3x + -1x2 = 3x + -3x + x2 + -1x2

Combine like terms: 3x + -3x = 0
7 + -3x + -1x2 = 0 + x2 + -1x2
7 + -3x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
7 + -3x + -1x2 = 0

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

Divide each side by '-1'.
-7 + 3x + x2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 3x + 7 + x2 = 0 + 7

Reorder the terms:
-7 + 7 + 3x + x2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 3x + x2 = 0 + 7
3x + x2 = 0 + 7

Combine like terms: 0 + 7 = 7
3x + x2 = 7

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 = 7 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 7 + 2.25

Combine like terms: 7 + 2.25 = 9.25
2.25 + 3x + x2 = 9.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(x + 1.5) equal to 3.041381265 and -3.041381265.

Subproblem 1

x + 1.5 = 3.041381265 Simplifying x + 1.5 = 3.041381265 Reorder the terms: 1.5 + x = 3.041381265 Solving 1.5 + x = 3.041381265 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 = 3.041381265 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 3.041381265 + -1.5 x = 3.041381265 + -1.5 Combine like terms: 3.041381265 + -1.5 = 1.541381265 x = 1.541381265 Simplifying x = 1.541381265

Subproblem 2

x + 1.5 = -3.041381265 Simplifying x + 1.5 = -3.041381265 Reorder the terms: 1.5 + x = -3.041381265 Solving 1.5 + x = -3.041381265 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 = -3.041381265 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -3.041381265 + -1.5 x = -3.041381265 + -1.5 Combine like terms: -3.041381265 + -1.5 = -4.541381265 x = -4.541381265 Simplifying x = -4.541381265

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.541381265, -4.541381265}

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