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Simplifying 3x(5x + -9) + 8(3x + -2) = 0 Reorder the terms: 3x(-9 + 5x) + 8(3x + -2) = 0 (-9 * 3x + 5x * 3x) + 8(3x + -2) = 0 (-27x + 15x2) + 8(3x + -2) = 0 Reorder the terms: -27x + 15x2 + 8(-2 + 3x) = 0 -27x + 15x2 + (-2 * 8 + 3x * 8) = 0 -27x + 15x2 + (-16 + 24x) = 0 Reorder the terms: -16 + -27x + 24x + 15x2 = 0 Combine like terms: -27x + 24x = -3x -16 + -3x + 15x2 = 0 Solving -16 + -3x + 15x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -1.066666667 + -0.2x + x2 = 0 Move the constant term to the right: Add '1.066666667' to each side of the equation. -1.066666667 + -0.2x + 1.066666667 + x2 = 0 + 1.066666667 Reorder the terms: -1.066666667 + 1.066666667 + -0.2x + x2 = 0 + 1.066666667 Combine like terms: -1.066666667 + 1.066666667 = 0.000000000 0.000000000 + -0.2x + x2 = 0 + 1.066666667 -0.2x + x2 = 0 + 1.066666667 Combine like terms: 0 + 1.066666667 = 1.066666667 -0.2x + x2 = 1.066666667 The x term is -0.2x. Take half its coefficient (-0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. -0.2x + 0.01 + x2 = 1.066666667 + 0.01 Reorder the terms: 0.01 + -0.2x + x2 = 1.066666667 + 0.01 Combine like terms: 1.066666667 + 0.01 = 1.076666667 0.01 + -0.2x + x2 = 1.076666667 Factor a perfect square on the left side: (x + -0.1)(x + -0.1) = 1.076666667 Calculate the square root of the right side: 1.037625495 Break this problem into two subproblems by setting (x + -0.1) equal to 1.037625495 and -1.037625495.Subproblem 1
x + -0.1 = 1.037625495 Simplifying x + -0.1 = 1.037625495 Reorder the terms: -0.1 + x = 1.037625495 Solving -0.1 + x = 1.037625495 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = 1.037625495 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = 1.037625495 + 0.1 x = 1.037625495 + 0.1 Combine like terms: 1.037625495 + 0.1 = 1.137625495 x = 1.137625495 Simplifying x = 1.137625495Subproblem 2
x + -0.1 = -1.037625495 Simplifying x + -0.1 = -1.037625495 Reorder the terms: -0.1 + x = -1.037625495 Solving -0.1 + x = -1.037625495 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = -1.037625495 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = -1.037625495 + 0.1 x = -1.037625495 + 0.1 Combine like terms: -1.037625495 + 0.1 = -0.937625495 x = -0.937625495 Simplifying x = -0.937625495Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.137625495, -0.937625495}
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