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Simplifying 0 = -9x2 + 4x + 3 Reorder the terms: 0 = 3 + 4x + -9x2 Solving 0 = 3 + 4x + -9x2 Solving for variable 'x'. Combine like terms: 0 + -3 = -3 -3 + -4x + 9x2 = 3 + 4x + -9x2 + -3 + -4x + 9x2 Reorder the terms: -3 + -4x + 9x2 = 3 + -3 + 4x + -4x + -9x2 + 9x2 Combine like terms: 3 + -3 = 0 -3 + -4x + 9x2 = 0 + 4x + -4x + -9x2 + 9x2 -3 + -4x + 9x2 = 4x + -4x + -9x2 + 9x2 Combine like terms: 4x + -4x = 0 -3 + -4x + 9x2 = 0 + -9x2 + 9x2 -3 + -4x + 9x2 = -9x2 + 9x2 Combine like terms: -9x2 + 9x2 = 0 -3 + -4x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.3333333333 + -0.4444444444x + x2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -0.4444444444x + 0.3333333333 + x2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -0.4444444444x + x2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -0.4444444444x + x2 = 0 + 0.3333333333 -0.4444444444x + x2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -0.4444444444x + x2 = 0.3333333333 The x term is -0.4444444444x. Take half its coefficient (-0.2222222222). Square it (0.04938271604) and add it to both sides. Add '0.04938271604' to each side of the equation. -0.4444444444x + 0.04938271604 + x2 = 0.3333333333 + 0.04938271604 Reorder the terms: 0.04938271604 + -0.4444444444x + x2 = 0.3333333333 + 0.04938271604 Combine like terms: 0.3333333333 + 0.04938271604 = 0.38271604934 0.04938271604 + -0.4444444444x + x2 = 0.38271604934 Factor a perfect square on the left side: (x + -0.2222222222)(x + -0.2222222222) = 0.38271604934 Calculate the square root of the right side: 0.618640485 Break this problem into two subproblems by setting (x + -0.2222222222) equal to 0.618640485 and -0.618640485.Subproblem 1
x + -0.2222222222 = 0.618640485 Simplifying x + -0.2222222222 = 0.618640485 Reorder the terms: -0.2222222222 + x = 0.618640485 Solving -0.2222222222 + x = 0.618640485 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + x = 0.618640485 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + x = 0.618640485 + 0.2222222222 x = 0.618640485 + 0.2222222222 Combine like terms: 0.618640485 + 0.2222222222 = 0.8408627072 x = 0.8408627072 Simplifying x = 0.8408627072Subproblem 2
x + -0.2222222222 = -0.618640485 Simplifying x + -0.2222222222 = -0.618640485 Reorder the terms: -0.2222222222 + x = -0.618640485 Solving -0.2222222222 + x = -0.618640485 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + x = -0.618640485 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + x = -0.618640485 + 0.2222222222 x = -0.618640485 + 0.2222222222 Combine like terms: -0.618640485 + 0.2222222222 = -0.3964182628 x = -0.3964182628 Simplifying x = -0.3964182628Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.8408627072, -0.3964182628}
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