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Simplifying 8x(2x + 7) = 11 Reorder the terms: 8x(7 + 2x) = 11 (7 * 8x + 2x * 8x) = 11 (56x + 16x2) = 11 Solving 56x + 16x2 = 11 Solving for variable 'x'. Reorder the terms: -11 + 56x + 16x2 = 11 + -11 Combine like terms: 11 + -11 = 0 -11 + 56x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.6875 + 3.5x + x2 = 0 Move the constant term to the right: Add '0.6875' to each side of the equation. -0.6875 + 3.5x + 0.6875 + x2 = 0 + 0.6875 Reorder the terms: -0.6875 + 0.6875 + 3.5x + x2 = 0 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + 3.5x + x2 = 0 + 0.6875 3.5x + x2 = 0 + 0.6875 Combine like terms: 0 + 0.6875 = 0.6875 3.5x + x2 = 0.6875 The x term is 3.5x. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5x + 3.0625 + x2 = 0.6875 + 3.0625 Reorder the terms: 3.0625 + 3.5x + x2 = 0.6875 + 3.0625 Combine like terms: 0.6875 + 3.0625 = 3.75 3.0625 + 3.5x + x2 = 3.75 Factor a perfect square on the left side: (x + 1.75)(x + 1.75) = 3.75 Calculate the square root of the right side: 1.936491673 Break this problem into two subproblems by setting (x + 1.75) equal to 1.936491673 and -1.936491673.Subproblem 1
x + 1.75 = 1.936491673 Simplifying x + 1.75 = 1.936491673 Reorder the terms: 1.75 + x = 1.936491673 Solving 1.75 + x = 1.936491673 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 1.936491673 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 1.936491673 + -1.75 x = 1.936491673 + -1.75 Combine like terms: 1.936491673 + -1.75 = 0.186491673 x = 0.186491673 Simplifying x = 0.186491673Subproblem 2
x + 1.75 = -1.936491673 Simplifying x + 1.75 = -1.936491673 Reorder the terms: 1.75 + x = -1.936491673 Solving 1.75 + x = -1.936491673 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -1.936491673 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -1.936491673 + -1.75 x = -1.936491673 + -1.75 Combine like terms: -1.936491673 + -1.75 = -3.686491673 x = -3.686491673 Simplifying x = -3.686491673Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.186491673, -3.686491673}
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