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Simplifying -16x2 + 40x + 3 = 10 Reorder the terms: 3 + 40x + -16x2 = 10 Solving 3 + 40x + -16x2 = 10 Solving for variable 'x'. Reorder the terms: 3 + -10 + 40x + -16x2 = 10 + -10 Combine like terms: 3 + -10 = -7 -7 + 40x + -16x2 = 10 + -10 Combine like terms: 10 + -10 = 0 -7 + 40x + -16x2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.4375 + -2.5x + x2 = 0 Move the constant term to the right: Add '-0.4375' to each side of the equation. 0.4375 + -2.5x + -0.4375 + x2 = 0 + -0.4375 Reorder the terms: 0.4375 + -0.4375 + -2.5x + x2 = 0 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + -2.5x + x2 = 0 + -0.4375 -2.5x + x2 = 0 + -0.4375 Combine like terms: 0 + -0.4375 = -0.4375 -2.5x + x2 = -0.4375 The x term is -2.5x. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5x + 1.5625 + x2 = -0.4375 + 1.5625 Reorder the terms: 1.5625 + -2.5x + x2 = -0.4375 + 1.5625 Combine like terms: -0.4375 + 1.5625 = 1.125 1.5625 + -2.5x + x2 = 1.125 Factor a perfect square on the left side: (x + -1.25)(x + -1.25) = 1.125 Calculate the square root of the right side: 1.060660172 Break this problem into two subproblems by setting (x + -1.25) equal to 1.060660172 and -1.060660172.Subproblem 1
x + -1.25 = 1.060660172 Simplifying x + -1.25 = 1.060660172 Reorder the terms: -1.25 + x = 1.060660172 Solving -1.25 + x = 1.060660172 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 1.060660172 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 1.060660172 + 1.25 x = 1.060660172 + 1.25 Combine like terms: 1.060660172 + 1.25 = 2.310660172 x = 2.310660172 Simplifying x = 2.310660172Subproblem 2
x + -1.25 = -1.060660172 Simplifying x + -1.25 = -1.060660172 Reorder the terms: -1.25 + x = -1.060660172 Solving -1.25 + x = -1.060660172 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -1.060660172 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -1.060660172 + 1.25 x = -1.060660172 + 1.25 Combine like terms: -1.060660172 + 1.25 = 0.189339828 x = 0.189339828 Simplifying x = 0.189339828Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.310660172, 0.189339828}
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