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Simplifying 9x2 + 22x + 10 = 0 Reorder the terms: 10 + 22x + 9x2 = 0 Solving 10 + 22x + 9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 1.111111111 + 2.444444444x + x2 = 0 Move the constant term to the right: Add '-1.111111111' to each side of the equation. 1.111111111 + 2.444444444x + -1.111111111 + x2 = 0 + -1.111111111 Reorder the terms: 1.111111111 + -1.111111111 + 2.444444444x + x2 = 0 + -1.111111111 Combine like terms: 1.111111111 + -1.111111111 = 0.000000000 0.000000000 + 2.444444444x + x2 = 0 + -1.111111111 2.444444444x + x2 = 0 + -1.111111111 Combine like terms: 0 + -1.111111111 = -1.111111111 2.444444444x + x2 = -1.111111111 The x term is 2.444444444x. Take half its coefficient (1.222222222). Square it (1.493827160) and add it to both sides. Add '1.493827160' to each side of the equation. 2.444444444x + 1.493827160 + x2 = -1.111111111 + 1.493827160 Reorder the terms: 1.493827160 + 2.444444444x + x2 = -1.111111111 + 1.493827160 Combine like terms: -1.111111111 + 1.493827160 = 0.382716049 1.493827160 + 2.444444444x + x2 = 0.382716049 Factor a perfect square on the left side: (x + 1.222222222)(x + 1.222222222) = 0.382716049 Calculate the square root of the right side: 0.618640484 Break this problem into two subproblems by setting (x + 1.222222222) equal to 0.618640484 and -0.618640484.Subproblem 1
x + 1.222222222 = 0.618640484 Simplifying x + 1.222222222 = 0.618640484 Reorder the terms: 1.222222222 + x = 0.618640484 Solving 1.222222222 + x = 0.618640484 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.222222222' to each side of the equation. 1.222222222 + -1.222222222 + x = 0.618640484 + -1.222222222 Combine like terms: 1.222222222 + -1.222222222 = 0.000000000 0.000000000 + x = 0.618640484 + -1.222222222 x = 0.618640484 + -1.222222222 Combine like terms: 0.618640484 + -1.222222222 = -0.603581738 x = -0.603581738 Simplifying x = -0.603581738Subproblem 2
x + 1.222222222 = -0.618640484 Simplifying x + 1.222222222 = -0.618640484 Reorder the terms: 1.222222222 + x = -0.618640484 Solving 1.222222222 + x = -0.618640484 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.222222222' to each side of the equation. 1.222222222 + -1.222222222 + x = -0.618640484 + -1.222222222 Combine like terms: 1.222222222 + -1.222222222 = 0.000000000 0.000000000 + x = -0.618640484 + -1.222222222 x = -0.618640484 + -1.222222222 Combine like terms: -0.618640484 + -1.222222222 = -1.840862706 x = -1.840862706 Simplifying x = -1.840862706Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.603581738, -1.840862706}
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