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Simplifying 10x + 24 = 8 + 8(x + 2) * 2x Reorder the terms: 24 + 10x = 8 + 8(x + 2) * 2x Reorder the terms: 24 + 10x = 8 + 8(2 + x) * 2x Reorder the terms for easier multiplication: 24 + 10x = 8 + 8 * 2x(2 + x) Multiply 8 * 2 24 + 10x = 8 + 16x(2 + x) 24 + 10x = 8 + (2 * 16x + x * 16x) 24 + 10x = 8 + (32x + 16x2) Solving 24 + 10x = 8 + 32x + 16x2 Solving for variable 'x'. Reorder the terms: 24 + -8 + 10x + -32x + -16x2 = 8 + 32x + 16x2 + -8 + -32x + -16x2 Combine like terms: 24 + -8 = 16 16 + 10x + -32x + -16x2 = 8 + 32x + 16x2 + -8 + -32x + -16x2 Combine like terms: 10x + -32x = -22x 16 + -22x + -16x2 = 8 + 32x + 16x2 + -8 + -32x + -16x2 Reorder the terms: 16 + -22x + -16x2 = 8 + -8 + 32x + -32x + 16x2 + -16x2 Combine like terms: 8 + -8 = 0 16 + -22x + -16x2 = 0 + 32x + -32x + 16x2 + -16x2 16 + -22x + -16x2 = 32x + -32x + 16x2 + -16x2 Combine like terms: 32x + -32x = 0 16 + -22x + -16x2 = 0 + 16x2 + -16x2 16 + -22x + -16x2 = 16x2 + -16x2 Combine like terms: 16x2 + -16x2 = 0 16 + -22x + -16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(8 + -11x + -8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(8 + -11x + -8x2)' equal to zero and attempt to solve: Simplifying 8 + -11x + -8x2 = 0 Solving 8 + -11x + -8x2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -1 + 1.375x + x2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 1.375x + 1 + x2 = 0 + 1 Reorder the terms: -1 + 1 + 1.375x + x2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 1.375x + x2 = 0 + 1 1.375x + x2 = 0 + 1 Combine like terms: 0 + 1 = 1 1.375x + x2 = 1 The x term is 1.375x. Take half its coefficient (0.6875). Square it (0.47265625) and add it to both sides. Add '0.47265625' to each side of the equation. 1.375x + 0.47265625 + x2 = 1 + 0.47265625 Reorder the terms: 0.47265625 + 1.375x + x2 = 1 + 0.47265625 Combine like terms: 1 + 0.47265625 = 1.47265625 0.47265625 + 1.375x + x2 = 1.47265625 Factor a perfect square on the left side: (x + 0.6875)(x + 0.6875) = 1.47265625 Calculate the square root of the right side: 1.21353049 Break this problem into two subproblems by setting (x + 0.6875) equal to 1.21353049 and -1.21353049.Subproblem 1
x + 0.6875 = 1.21353049 Simplifying x + 0.6875 = 1.21353049 Reorder the terms: 0.6875 + x = 1.21353049 Solving 0.6875 + x = 1.21353049 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + x = 1.21353049 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + x = 1.21353049 + -0.6875 x = 1.21353049 + -0.6875 Combine like terms: 1.21353049 + -0.6875 = 0.52603049 x = 0.52603049 Simplifying x = 0.52603049Subproblem 2
x + 0.6875 = -1.21353049 Simplifying x + 0.6875 = -1.21353049 Reorder the terms: 0.6875 + x = -1.21353049 Solving 0.6875 + x = -1.21353049 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + x = -1.21353049 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + x = -1.21353049 + -0.6875 x = -1.21353049 + -0.6875 Combine like terms: -1.21353049 + -0.6875 = -1.90103049 x = -1.90103049 Simplifying x = -1.90103049Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.52603049, -1.90103049}Solution
x = {0.52603049, -1.90103049}
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