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Simplifying 0 = 9x2 + -22x + 10 Reorder the terms: 0 = 10 + -22x + 9x2 Solving 0 = 10 + -22x + 9x2 Solving for variable 'x'. Combine like terms: 0 + -10 = -10 -10 + 22x + -9x2 = 10 + -22x + 9x2 + -10 + 22x + -9x2 Reorder the terms: -10 + 22x + -9x2 = 10 + -10 + -22x + 22x + 9x2 + -9x2 Combine like terms: 10 + -10 = 0 -10 + 22x + -9x2 = 0 + -22x + 22x + 9x2 + -9x2 -10 + 22x + -9x2 = -22x + 22x + 9x2 + -9x2 Combine like terms: -22x + 22x = 0 -10 + 22x + -9x2 = 0 + 9x2 + -9x2 -10 + 22x + -9x2 = 9x2 + -9x2 Combine like terms: 9x2 + -9x2 = 0 -10 + 22x + -9x2 = 0 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 = 1.840862706 x = 1.840862706 Simplifying x = 1.840862706Subproblem 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 = 0.603581738 x = 0.603581738 Simplifying x = 0.603581738Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.840862706, 0.603581738}
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