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Simplifying 8x(2x + 3) = -1 Reorder the terms: 8x(3 + 2x) = -1 (3 * 8x + 2x * 8x) = -1 (24x + 16x2) = -1 Solving 24x + 16x2 = -1 Solving for variable 'x'. Reorder the terms: 1 + 24x + 16x2 = -1 + 1 Combine like terms: -1 + 1 = 0 1 + 24x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.0625 + 1.5x + x2 = 0 Move the constant term to the right: Add '-0.0625' to each side of the equation. 0.0625 + 1.5x + -0.0625 + x2 = 0 + -0.0625 Reorder the terms: 0.0625 + -0.0625 + 1.5x + x2 = 0 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + 1.5x + x2 = 0 + -0.0625 1.5x + x2 = 0 + -0.0625 Combine like terms: 0 + -0.0625 = -0.0625 1.5x + x2 = -0.0625 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 = -0.0625 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = -0.0625 + 0.5625 Combine like terms: -0.0625 + 0.5625 = 0.5 0.5625 + 1.5x + x2 = 0.5 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (x + 0.75) equal to 0.707106781 and -0.707106781.Subproblem 1
x + 0.75 = 0.707106781 Simplifying x + 0.75 = 0.707106781 Reorder the terms: 0.75 + x = 0.707106781 Solving 0.75 + x = 0.707106781 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 = 0.707106781 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 0.707106781 + -0.75 x = 0.707106781 + -0.75 Combine like terms: 0.707106781 + -0.75 = -0.042893219 x = -0.042893219 Simplifying x = -0.042893219Subproblem 2
x + 0.75 = -0.707106781 Simplifying x + 0.75 = -0.707106781 Reorder the terms: 0.75 + x = -0.707106781 Solving 0.75 + x = -0.707106781 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 = -0.707106781 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -0.707106781 + -0.75 x = -0.707106781 + -0.75 Combine like terms: -0.707106781 + -0.75 = -1.457106781 x = -1.457106781 Simplifying x = -1.457106781Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.042893219, -1.457106781}
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