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Simplifying 4x2 + 12.5x + 9.75 = 0 Reorder the terms: 9.75 + 12.5x + 4x2 = 0 Solving 9.75 + 12.5x + 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'. 2.4375 + 3.125x + x2 = 0 Move the constant term to the right: Add '-2.4375' to each side of the equation. 2.4375 + 3.125x + -2.4375 + x2 = 0 + -2.4375 Reorder the terms: 2.4375 + -2.4375 + 3.125x + x2 = 0 + -2.4375 Combine like terms: 2.4375 + -2.4375 = 0.0000 0.0000 + 3.125x + x2 = 0 + -2.4375 3.125x + x2 = 0 + -2.4375 Combine like terms: 0 + -2.4375 = -2.4375 3.125x + x2 = -2.4375 The x term is 3.125x. Take half its coefficient (1.5625). Square it (2.44140625) and add it to both sides. Add '2.44140625' to each side of the equation. 3.125x + 2.44140625 + x2 = -2.4375 + 2.44140625 Reorder the terms: 2.44140625 + 3.125x + x2 = -2.4375 + 2.44140625 Combine like terms: -2.4375 + 2.44140625 = 0.00390625 2.44140625 + 3.125x + x2 = 0.00390625 Factor a perfect square on the left side: (x + 1.5625)(x + 1.5625) = 0.00390625 Calculate the square root of the right side: 0.0625 Break this problem into two subproblems by setting (x + 1.5625) equal to 0.0625 and -0.0625.Subproblem 1
x + 1.5625 = 0.0625 Simplifying x + 1.5625 = 0.0625 Reorder the terms: 1.5625 + x = 0.0625 Solving 1.5625 + x = 0.0625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5625' to each side of the equation. 1.5625 + -1.5625 + x = 0.0625 + -1.5625 Combine like terms: 1.5625 + -1.5625 = 0.0000 0.0000 + x = 0.0625 + -1.5625 x = 0.0625 + -1.5625 Combine like terms: 0.0625 + -1.5625 = -1.5 x = -1.5 Simplifying x = -1.5Subproblem 2
x + 1.5625 = -0.0625 Simplifying x + 1.5625 = -0.0625 Reorder the terms: 1.5625 + x = -0.0625 Solving 1.5625 + x = -0.0625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5625' to each side of the equation. 1.5625 + -1.5625 + x = -0.0625 + -1.5625 Combine like terms: 1.5625 + -1.5625 = 0.0000 0.0000 + x = -0.0625 + -1.5625 x = -0.0625 + -1.5625 Combine like terms: -0.0625 + -1.5625 = -1.625 x = -1.625 Simplifying x = -1.625Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.5, -1.625}
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