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Simplifying 4x2 + 7x + 2.5 = 0 Reorder the terms: 2.5 + 7x + 4x2 = 0 Solving 2.5 + 7x + 4x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.625 + 1.75x + x2 = 0 Move the constant term to the right: Add '-0.625' to each side of the equation. 0.625 + 1.75x + -0.625 + x2 = 0 + -0.625 Reorder the terms: 0.625 + -0.625 + 1.75x + x2 = 0 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + 1.75x + x2 = 0 + -0.625 1.75x + x2 = 0 + -0.625 Combine like terms: 0 + -0.625 = -0.625 1.75x + x2 = -0.625 The x term is 1.75x. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75x + 0.765625 + x2 = -0.625 + 0.765625 Reorder the terms: 0.765625 + 1.75x + x2 = -0.625 + 0.765625 Combine like terms: -0.625 + 0.765625 = 0.140625 0.765625 + 1.75x + x2 = 0.140625 Factor a perfect square on the left side: (x + 0.875)(x + 0.875) = 0.140625 Calculate the square root of the right side: 0.375 Break this problem into two subproblems by setting (x + 0.875) equal to 0.375 and -0.375.Subproblem 1
x + 0.875 = 0.375 Simplifying x + 0.875 = 0.375 Reorder the terms: 0.875 + x = 0.375 Solving 0.875 + x = 0.375 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = 0.375 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = 0.375 + -0.875 x = 0.375 + -0.875 Combine like terms: 0.375 + -0.875 = -0.5 x = -0.5 Simplifying x = -0.5Subproblem 2
x + 0.875 = -0.375 Simplifying x + 0.875 = -0.375 Reorder the terms: 0.875 + x = -0.375 Solving 0.875 + x = -0.375 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = -0.375 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = -0.375 + -0.875 x = -0.375 + -0.875 Combine like terms: -0.375 + -0.875 = -1.25 x = -1.25 Simplifying x = -1.25Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.5, -1.25}
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