10x^2=22-3x

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Solution for 10x^2=22-3x equation:


Simplifying
10x2 = 22 + -3x

Solving
10x2 = 22 + -3x

Solving for variable 'x'.

Reorder the terms:
-22 + 3x + 10x2 = 22 + -3x + -22 + 3x

Reorder the terms:
-22 + 3x + 10x2 = 22 + -22 + -3x + 3x

Combine like terms: 22 + -22 = 0
-22 + 3x + 10x2 = 0 + -3x + 3x
-22 + 3x + 10x2 = -3x + 3x

Combine like terms: -3x + 3x = 0
-22 + 3x + 10x2 = 0

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

Divide each side by '10'.
-2.2 + 0.3x + x2 = 0

Move the constant term to the right:

Add '2.2' to each side of the equation.
-2.2 + 0.3x + 2.2 + x2 = 0 + 2.2

Reorder the terms:
-2.2 + 2.2 + 0.3x + x2 = 0 + 2.2

Combine like terms: -2.2 + 2.2 = 0.0
0.0 + 0.3x + x2 = 0 + 2.2
0.3x + x2 = 0 + 2.2

Combine like terms: 0 + 2.2 = 2.2
0.3x + x2 = 2.2

The x term is 0.3x.  Take half its coefficient (0.15).
Square it (0.0225) and add it to both sides.

Add '0.0225' to each side of the equation.
0.3x + 0.0225 + x2 = 2.2 + 0.0225

Reorder the terms:
0.0225 + 0.3x + x2 = 2.2 + 0.0225

Combine like terms: 2.2 + 0.0225 = 2.2225
0.0225 + 0.3x + x2 = 2.2225

Factor a perfect square on the left side:
(x + 0.15)(x + 0.15) = 2.2225

Calculate the square root of the right side: 1.490805152

Break this problem into two subproblems by setting 
(x + 0.15) equal to 1.490805152 and -1.490805152.

Subproblem 1

x + 0.15 = 1.490805152 Simplifying x + 0.15 = 1.490805152 Reorder the terms: 0.15 + x = 1.490805152 Solving 0.15 + x = 1.490805152 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + x = 1.490805152 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + x = 1.490805152 + -0.15 x = 1.490805152 + -0.15 Combine like terms: 1.490805152 + -0.15 = 1.340805152 x = 1.340805152 Simplifying x = 1.340805152

Subproblem 2

x + 0.15 = -1.490805152 Simplifying x + 0.15 = -1.490805152 Reorder the terms: 0.15 + x = -1.490805152 Solving 0.15 + x = -1.490805152 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + x = -1.490805152 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + x = -1.490805152 + -0.15 x = -1.490805152 + -0.15 Combine like terms: -1.490805152 + -0.15 = -1.640805152 x = -1.640805152 Simplifying x = -1.640805152

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.340805152, -1.640805152}

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