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Simplifying 3(2x2 + -3x + 5) + -5(3x2 + -4x + -6) = 0 Reorder the terms: 3(5 + -3x + 2x2) + -5(3x2 + -4x + -6) = 0 (5 * 3 + -3x * 3 + 2x2 * 3) + -5(3x2 + -4x + -6) = 0 (15 + -9x + 6x2) + -5(3x2 + -4x + -6) = 0 Reorder the terms: 15 + -9x + 6x2 + -5(-6 + -4x + 3x2) = 0 15 + -9x + 6x2 + (-6 * -5 + -4x * -5 + 3x2 * -5) = 0 15 + -9x + 6x2 + (30 + 20x + -15x2) = 0 Reorder the terms: 15 + 30 + -9x + 20x + 6x2 + -15x2 = 0 Combine like terms: 15 + 30 = 45 45 + -9x + 20x + 6x2 + -15x2 = 0 Combine like terms: -9x + 20x = 11x 45 + 11x + 6x2 + -15x2 = 0 Combine like terms: 6x2 + -15x2 = -9x2 45 + 11x + -9x2 = 0 Solving 45 + 11x + -9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -5 + -1.222222222x + x2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -1.222222222x + 5 + x2 = 0 + 5 Reorder the terms: -5 + 5 + -1.222222222x + x2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -1.222222222x + x2 = 0 + 5 -1.222222222x + x2 = 0 + 5 Combine like terms: 0 + 5 = 5 -1.222222222x + x2 = 5 The x term is -1.222222222x. Take half its coefficient (-0.611111111). Square it (0.3734567900) and add it to both sides. Add '0.3734567900' to each side of the equation. -1.222222222x + 0.3734567900 + x2 = 5 + 0.3734567900 Reorder the terms: 0.3734567900 + -1.222222222x + x2 = 5 + 0.3734567900 Combine like terms: 5 + 0.3734567900 = 5.37345679 0.3734567900 + -1.222222222x + x2 = 5.37345679 Factor a perfect square on the left side: (x + -0.611111111)(x + -0.611111111) = 5.37345679 Calculate the square root of the right side: 2.318071783 Break this problem into two subproblems by setting (x + -0.611111111) equal to 2.318071783 and -2.318071783.Subproblem 1
x + -0.611111111 = 2.318071783 Simplifying x + -0.611111111 = 2.318071783 Reorder the terms: -0.611111111 + x = 2.318071783 Solving -0.611111111 + x = 2.318071783 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.611111111' to each side of the equation. -0.611111111 + 0.611111111 + x = 2.318071783 + 0.611111111 Combine like terms: -0.611111111 + 0.611111111 = 0.000000000 0.000000000 + x = 2.318071783 + 0.611111111 x = 2.318071783 + 0.611111111 Combine like terms: 2.318071783 + 0.611111111 = 2.929182894 x = 2.929182894 Simplifying x = 2.929182894Subproblem 2
x + -0.611111111 = -2.318071783 Simplifying x + -0.611111111 = -2.318071783 Reorder the terms: -0.611111111 + x = -2.318071783 Solving -0.611111111 + x = -2.318071783 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.611111111' to each side of the equation. -0.611111111 + 0.611111111 + x = -2.318071783 + 0.611111111 Combine like terms: -0.611111111 + 0.611111111 = 0.000000000 0.000000000 + x = -2.318071783 + 0.611111111 x = -2.318071783 + 0.611111111 Combine like terms: -2.318071783 + 0.611111111 = -1.706960672 x = -1.706960672 Simplifying x = -1.706960672Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.929182894, -1.706960672}
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