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Simplifying 16x2 = 24x + 19 Reorder the terms: 16x2 = 19 + 24x Solving 16x2 = 19 + 24x Solving for variable 'x'. Reorder the terms: -19 + -24x + 16x2 = 19 + 24x + -19 + -24x Reorder the terms: -19 + -24x + 16x2 = 19 + -19 + 24x + -24x Combine like terms: 19 + -19 = 0 -19 + -24x + 16x2 = 0 + 24x + -24x -19 + -24x + 16x2 = 24x + -24x Combine like terms: 24x + -24x = 0 -19 + -24x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.1875 + -1.5x + x2 = 0 Move the constant term to the right: Add '1.1875' to each side of the equation. -1.1875 + -1.5x + 1.1875 + x2 = 0 + 1.1875 Reorder the terms: -1.1875 + 1.1875 + -1.5x + x2 = 0 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + -1.5x + x2 = 0 + 1.1875 -1.5x + x2 = 0 + 1.1875 Combine like terms: 0 + 1.1875 = 1.1875 -1.5x + x2 = 1.1875 The x term is -1.5x. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5x + 0.5625 + x2 = 1.1875 + 0.5625 Reorder the terms: 0.5625 + -1.5x + x2 = 1.1875 + 0.5625 Combine like terms: 1.1875 + 0.5625 = 1.75 0.5625 + -1.5x + x2 = 1.75 Factor a perfect square on the left side: (x + -0.75)(x + -0.75) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (x + -0.75) equal to 1.322875656 and -1.322875656.Subproblem 1
x + -0.75 = 1.322875656 Simplifying x + -0.75 = 1.322875656 Reorder the terms: -0.75 + x = 1.322875656 Solving -0.75 + x = 1.322875656 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 1.322875656 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 1.322875656 + 0.75 x = 1.322875656 + 0.75 Combine like terms: 1.322875656 + 0.75 = 2.072875656 x = 2.072875656 Simplifying x = 2.072875656Subproblem 2
x + -0.75 = -1.322875656 Simplifying x + -0.75 = -1.322875656 Reorder the terms: -0.75 + x = -1.322875656 Solving -0.75 + x = -1.322875656 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -1.322875656 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -1.322875656 + 0.75 x = -1.322875656 + 0.75 Combine like terms: -1.322875656 + 0.75 = -0.572875656 x = -0.572875656 Simplifying x = -0.572875656Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.072875656, -0.572875656}
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