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Simplifying -16x2 + 40x + -19 = 0 Reorder the terms: -19 + 40x + -16x2 = 0 Solving -19 + 40x + -16x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 1.1875 + -2.5x + x2 = 0 Move the constant term to the right: Add '-1.1875' to each side of the equation. 1.1875 + -2.5x + -1.1875 + x2 = 0 + -1.1875 Reorder the terms: 1.1875 + -1.1875 + -2.5x + x2 = 0 + -1.1875 Combine like terms: 1.1875 + -1.1875 = 0.0000 0.0000 + -2.5x + x2 = 0 + -1.1875 -2.5x + x2 = 0 + -1.1875 Combine like terms: 0 + -1.1875 = -1.1875 -2.5x + x2 = -1.1875 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 = -1.1875 + 1.5625 Reorder the terms: 1.5625 + -2.5x + x2 = -1.1875 + 1.5625 Combine like terms: -1.1875 + 1.5625 = 0.375 1.5625 + -2.5x + x2 = 0.375 Factor a perfect square on the left side: (x + -1.25)(x + -1.25) = 0.375 Calculate the square root of the right side: 0.612372436 Break this problem into two subproblems by setting (x + -1.25) equal to 0.612372436 and -0.612372436.Subproblem 1
x + -1.25 = 0.612372436 Simplifying x + -1.25 = 0.612372436 Reorder the terms: -1.25 + x = 0.612372436 Solving -1.25 + x = 0.612372436 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 = 0.612372436 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 0.612372436 + 1.25 x = 0.612372436 + 1.25 Combine like terms: 0.612372436 + 1.25 = 1.862372436 x = 1.862372436 Simplifying x = 1.862372436Subproblem 2
x + -1.25 = -0.612372436 Simplifying x + -1.25 = -0.612372436 Reorder the terms: -1.25 + x = -0.612372436 Solving -1.25 + x = -0.612372436 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 = -0.612372436 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -0.612372436 + 1.25 x = -0.612372436 + 1.25 Combine like terms: -0.612372436 + 1.25 = 0.637627564 x = 0.637627564 Simplifying x = 0.637627564Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.862372436, 0.637627564}
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