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Simplifying 10 = -16x2 + 24x + 3 Reorder the terms: 10 = 3 + 24x + -16x2 Solving 10 = 3 + 24x + -16x2 Solving for variable 'x'. Combine like terms: 10 + -3 = 7 7 + -24x + 16x2 = 3 + 24x + -16x2 + -3 + -24x + 16x2 Reorder the terms: 7 + -24x + 16x2 = 3 + -3 + 24x + -24x + -16x2 + 16x2 Combine like terms: 3 + -3 = 0 7 + -24x + 16x2 = 0 + 24x + -24x + -16x2 + 16x2 7 + -24x + 16x2 = 24x + -24x + -16x2 + 16x2 Combine like terms: 24x + -24x = 0 7 + -24x + 16x2 = 0 + -16x2 + 16x2 7 + -24x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 7 + -24x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.4375 + -1.5x + x2 = 0 Move the constant term to the right: Add '-0.4375' to each side of the equation. 0.4375 + -1.5x + -0.4375 + x2 = 0 + -0.4375 Reorder the terms: 0.4375 + -0.4375 + -1.5x + x2 = 0 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + -1.5x + x2 = 0 + -0.4375 -1.5x + x2 = 0 + -0.4375 Combine like terms: 0 + -0.4375 = -0.4375 -1.5x + x2 = -0.4375 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 = -0.4375 + 0.5625 Reorder the terms: 0.5625 + -1.5x + x2 = -0.4375 + 0.5625 Combine like terms: -0.4375 + 0.5625 = 0.125 0.5625 + -1.5x + x2 = 0.125 Factor a perfect square on the left side: (x + -0.75)(x + -0.75) = 0.125 Calculate the square root of the right side: 0.353553391 Break this problem into two subproblems by setting (x + -0.75) equal to 0.353553391 and -0.353553391.Subproblem 1
x + -0.75 = 0.353553391 Simplifying x + -0.75 = 0.353553391 Reorder the terms: -0.75 + x = 0.353553391 Solving -0.75 + x = 0.353553391 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 = 0.353553391 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 0.353553391 + 0.75 x = 0.353553391 + 0.75 Combine like terms: 0.353553391 + 0.75 = 1.103553391 x = 1.103553391 Simplifying x = 1.103553391Subproblem 2
x + -0.75 = -0.353553391 Simplifying x + -0.75 = -0.353553391 Reorder the terms: -0.75 + x = -0.353553391 Solving -0.75 + x = -0.353553391 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 = -0.353553391 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -0.353553391 + 0.75 x = -0.353553391 + 0.75 Combine like terms: -0.353553391 + 0.75 = 0.396446609 x = 0.396446609 Simplifying x = 0.396446609Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.103553391, 0.396446609}
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