22x^2+5x+3=15

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Solution for 22x^2+5x+3=15 equation:


Simplifying
22x2 + 5x + 3 = 15

Reorder the terms:
3 + 5x + 22x2 = 15

Solving
3 + 5x + 22x2 = 15

Solving for variable 'x'.

Reorder the terms:
3 + -15 + 5x + 22x2 = 15 + -15

Combine like terms: 3 + -15 = -12
-12 + 5x + 22x2 = 15 + -15

Combine like terms: 15 + -15 = 0
-12 + 5x + 22x2 = 0

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

Divide each side by '22'.
-0.5454545455 + 0.2272727273x + x2 = 0

Move the constant term to the right:

Add '0.5454545455' to each side of the equation.
-0.5454545455 + 0.2272727273x + 0.5454545455 + x2 = 0 + 0.5454545455

Reorder the terms:
-0.5454545455 + 0.5454545455 + 0.2272727273x + x2 = 0 + 0.5454545455

Combine like terms: -0.5454545455 + 0.5454545455 = 0.0000000000
0.0000000000 + 0.2272727273x + x2 = 0 + 0.5454545455
0.2272727273x + x2 = 0 + 0.5454545455

Combine like terms: 0 + 0.5454545455 = 0.5454545455
0.2272727273x + x2 = 0.5454545455

The x term is 0.2272727273x.  Take half its coefficient (0.1136363637).
Square it (0.01291322316) and add it to both sides.

Add '0.01291322316' to each side of the equation.
0.2272727273x + 0.01291322316 + x2 = 0.5454545455 + 0.01291322316

Reorder the terms:
0.01291322316 + 0.2272727273x + x2 = 0.5454545455 + 0.01291322316

Combine like terms: 0.5454545455 + 0.01291322316 = 0.55836776866
0.01291322316 + 0.2272727273x + x2 = 0.55836776866

Factor a perfect square on the left side:
(x + 0.1136363637)(x + 0.1136363637) = 0.55836776866

Calculate the square root of the right side: 0.747240101

Break this problem into two subproblems by setting 
(x + 0.1136363637) equal to 0.747240101 and -0.747240101.

Subproblem 1

x + 0.1136363637 = 0.747240101 Simplifying x + 0.1136363637 = 0.747240101 Reorder the terms: 0.1136363637 + x = 0.747240101 Solving 0.1136363637 + x = 0.747240101 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1136363637' to each side of the equation. 0.1136363637 + -0.1136363637 + x = 0.747240101 + -0.1136363637 Combine like terms: 0.1136363637 + -0.1136363637 = 0.0000000000 0.0000000000 + x = 0.747240101 + -0.1136363637 x = 0.747240101 + -0.1136363637 Combine like terms: 0.747240101 + -0.1136363637 = 0.6336037373 x = 0.6336037373 Simplifying x = 0.6336037373

Subproblem 2

x + 0.1136363637 = -0.747240101 Simplifying x + 0.1136363637 = -0.747240101 Reorder the terms: 0.1136363637 + x = -0.747240101 Solving 0.1136363637 + x = -0.747240101 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1136363637' to each side of the equation. 0.1136363637 + -0.1136363637 + x = -0.747240101 + -0.1136363637 Combine like terms: 0.1136363637 + -0.1136363637 = 0.0000000000 0.0000000000 + x = -0.747240101 + -0.1136363637 x = -0.747240101 + -0.1136363637 Combine like terms: -0.747240101 + -0.1136363637 = -0.8608764647 x = -0.8608764647 Simplifying x = -0.8608764647

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.6336037373, -0.8608764647}

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