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Simplifying 1.5 = -2x2 + 4x Reorder the terms: 1.5 = 4x + -2x2 Solving 1.5 = 4x + -2x2 Solving for variable 'x'. Reorder the terms: 1.5 + -4x + 2x2 = 4x + -4x + -2x2 + 2x2 Combine like terms: 4x + -4x = 0 1.5 + -4x + 2x2 = 0 + -2x2 + 2x2 1.5 + -4x + 2x2 = -2x2 + 2x2 Combine like terms: -2x2 + 2x2 = 0 1.5 + -4x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.75 + -2x + x2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + -2x + -0.75 + x2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + -2x + x2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + -2x + x2 = 0 + -0.75 -2x + x2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 -2x + x2 = -0.75 The x term is -2x. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2x + 1 + x2 = -0.75 + 1 Reorder the terms: 1 + -2x + x2 = -0.75 + 1 Combine like terms: -0.75 + 1 = 0.25 1 + -2x + x2 = 0.25 Factor a perfect square on the left side: (x + -1)(x + -1) = 0.25 Calculate the square root of the right side: 0.5 Break this problem into two subproblems by setting (x + -1) equal to 0.5 and -0.5.Subproblem 1
x + -1 = 0.5 Simplifying x + -1 = 0.5 Reorder the terms: -1 + x = 0.5 Solving -1 + x = 0.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 0.5 + 1 Combine like terms: -1 + 1 = 0 0 + x = 0.5 + 1 x = 0.5 + 1 Combine like terms: 0.5 + 1 = 1.5 x = 1.5 Simplifying x = 1.5Subproblem 2
x + -1 = -0.5 Simplifying x + -1 = -0.5 Reorder the terms: -1 + x = -0.5 Solving -1 + x = -0.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -0.5 + 1 Combine like terms: -1 + 1 = 0 0 + x = -0.5 + 1 x = -0.5 + 1 Combine like terms: -0.5 + 1 = 0.5 x = 0.5 Simplifying x = 0.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.5, 0.5}
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