0=5x^2+3.5x-6

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Solution for 0=5x^2+3.5x-6 equation:


Simplifying
0 = 5x2 + 3.5x + -6

Reorder the terms:
0 = -6 + 3.5x + 5x2

Solving
0 = -6 + 3.5x + 5x2

Solving for variable 'x'.

Combine like terms: 0 + 6 = 6
6 + -3.5x + -5x2 = -6 + 3.5x + 5x2 + 6 + -3.5x + -5x2

Reorder the terms:
6 + -3.5x + -5x2 = -6 + 6 + 3.5x + -3.5x + 5x2 + -5x2

Combine like terms: -6 + 6 = 0
6 + -3.5x + -5x2 = 0 + 3.5x + -3.5x + 5x2 + -5x2
6 + -3.5x + -5x2 = 3.5x + -3.5x + 5x2 + -5x2

Combine like terms: 3.5x + -3.5x = 0.0
6 + -3.5x + -5x2 = 0.0 + 5x2 + -5x2
6 + -3.5x + -5x2 = 5x2 + -5x2

Combine like terms: 5x2 + -5x2 = 0
6 + -3.5x + -5x2 = 0

Begin completing the square.  Divide all terms by
-5 the coefficient of the squared term: 

Divide each side by '-5'.
-1.2 + 0.7x + x2 = 0

Move the constant term to the right:

Add '1.2' to each side of the equation.
-1.2 + 0.7x + 1.2 + x2 = 0 + 1.2

Reorder the terms:
-1.2 + 1.2 + 0.7x + x2 = 0 + 1.2

Combine like terms: -1.2 + 1.2 = 0.0
0.0 + 0.7x + x2 = 0 + 1.2
0.7x + x2 = 0 + 1.2

Combine like terms: 0 + 1.2 = 1.2
0.7x + x2 = 1.2

The x term is 0.7x.  Take half its coefficient (0.35).
Square it (0.1225) and add it to both sides.

Add '0.1225' to each side of the equation.
0.7x + 0.1225 + x2 = 1.2 + 0.1225

Reorder the terms:
0.1225 + 0.7x + x2 = 1.2 + 0.1225

Combine like terms: 1.2 + 0.1225 = 1.3225
0.1225 + 0.7x + x2 = 1.3225

Factor a perfect square on the left side:
(x + 0.35)(x + 0.35) = 1.3225

Calculate the square root of the right side: 1.15

Break this problem into two subproblems by setting 
(x + 0.35) equal to 1.15 and -1.15.

Subproblem 1

x + 0.35 = 1.15 Simplifying x + 0.35 = 1.15 Reorder the terms: 0.35 + x = 1.15 Solving 0.35 + x = 1.15 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.35' to each side of the equation. 0.35 + -0.35 + x = 1.15 + -0.35 Combine like terms: 0.35 + -0.35 = 0.00 0.00 + x = 1.15 + -0.35 x = 1.15 + -0.35 Combine like terms: 1.15 + -0.35 = 0.8 x = 0.8 Simplifying x = 0.8

Subproblem 2

x + 0.35 = -1.15 Simplifying x + 0.35 = -1.15 Reorder the terms: 0.35 + x = -1.15 Solving 0.35 + x = -1.15 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.35' to each side of the equation. 0.35 + -0.35 + x = -1.15 + -0.35 Combine like terms: 0.35 + -0.35 = 0.00 0.00 + x = -1.15 + -0.35 x = -1.15 + -0.35 Combine like terms: -1.15 + -0.35 = -1.5 x = -1.5 Simplifying x = -1.5

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.8, -1.5}

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