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Simplifying 4(6 + x) = x(8x + 7) (6 * 4 + x * 4) = x(8x + 7) (24 + 4x) = x(8x + 7) Reorder the terms: 24 + 4x = x(7 + 8x) 24 + 4x = (7 * x + 8x * x) 24 + 4x = (7x + 8x2) Solving 24 + 4x = 7x + 8x2 Solving for variable 'x'. Combine like terms: 4x + -7x = -3x 24 + -3x + -8x2 = 7x + 8x2 + -7x + -8x2 Reorder the terms: 24 + -3x + -8x2 = 7x + -7x + 8x2 + -8x2 Combine like terms: 7x + -7x = 0 24 + -3x + -8x2 = 0 + 8x2 + -8x2 24 + -3x + -8x2 = 8x2 + -8x2 Combine like terms: 8x2 + -8x2 = 0 24 + -3x + -8x2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -3 + 0.375x + x2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 0.375x + 3 + x2 = 0 + 3 Reorder the terms: -3 + 3 + 0.375x + x2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 0.375x + x2 = 0 + 3 0.375x + x2 = 0 + 3 Combine like terms: 0 + 3 = 3 0.375x + x2 = 3 The x term is 0.375x. Take half its coefficient (0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. 0.375x + 0.03515625 + x2 = 3 + 0.03515625 Reorder the terms: 0.03515625 + 0.375x + x2 = 3 + 0.03515625 Combine like terms: 3 + 0.03515625 = 3.03515625 0.03515625 + 0.375x + x2 = 3.03515625 Factor a perfect square on the left side: (x + 0.1875)(x + 0.1875) = 3.03515625 Calculate the square root of the right side: 1.742169983 Break this problem into two subproblems by setting (x + 0.1875) equal to 1.742169983 and -1.742169983.Subproblem 1
x + 0.1875 = 1.742169983 Simplifying x + 0.1875 = 1.742169983 Reorder the terms: 0.1875 + x = 1.742169983 Solving 0.1875 + x = 1.742169983 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + x = 1.742169983 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + x = 1.742169983 + -0.1875 x = 1.742169983 + -0.1875 Combine like terms: 1.742169983 + -0.1875 = 1.554669983 x = 1.554669983 Simplifying x = 1.554669983Subproblem 2
x + 0.1875 = -1.742169983 Simplifying x + 0.1875 = -1.742169983 Reorder the terms: 0.1875 + x = -1.742169983 Solving 0.1875 + x = -1.742169983 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + x = -1.742169983 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + x = -1.742169983 + -0.1875 x = -1.742169983 + -0.1875 Combine like terms: -1.742169983 + -0.1875 = -1.929669983 x = -1.929669983 Simplifying x = -1.929669983Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.554669983, -1.929669983}
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