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Simplifying (5x2 + -7x + 14) + (8x2 + 9x + -69) = 0 Reorder the terms: (14 + -7x + 5x2) + (8x2 + 9x + -69) = 0 Remove parenthesis around (14 + -7x + 5x2) 14 + -7x + 5x2 + (8x2 + 9x + -69) = 0 Reorder the terms: 14 + -7x + 5x2 + (-69 + 9x + 8x2) = 0 Remove parenthesis around (-69 + 9x + 8x2) 14 + -7x + 5x2 + -69 + 9x + 8x2 = 0 Reorder the terms: 14 + -69 + -7x + 9x + 5x2 + 8x2 = 0 Combine like terms: 14 + -69 = -55 -55 + -7x + 9x + 5x2 + 8x2 = 0 Combine like terms: -7x + 9x = 2x -55 + 2x + 5x2 + 8x2 = 0 Combine like terms: 5x2 + 8x2 = 13x2 -55 + 2x + 13x2 = 0 Solving -55 + 2x + 13x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -4.230769231 + 0.1538461538x + x2 = 0 Move the constant term to the right: Add '4.230769231' to each side of the equation. -4.230769231 + 0.1538461538x + 4.230769231 + x2 = 0 + 4.230769231 Reorder the terms: -4.230769231 + 4.230769231 + 0.1538461538x + x2 = 0 + 4.230769231 Combine like terms: -4.230769231 + 4.230769231 = 0.000000000 0.000000000 + 0.1538461538x + x2 = 0 + 4.230769231 0.1538461538x + x2 = 0 + 4.230769231 Combine like terms: 0 + 4.230769231 = 4.230769231 0.1538461538x + x2 = 4.230769231 The x term is 0.1538461538x. Take half its coefficient (0.0769230769). Square it (0.005917159760) and add it to both sides. Add '0.005917159760' to each side of the equation. 0.1538461538x + 0.005917159760 + x2 = 4.230769231 + 0.005917159760 Reorder the terms: 0.005917159760 + 0.1538461538x + x2 = 4.230769231 + 0.005917159760 Combine like terms: 4.230769231 + 0.005917159760 = 4.23668639076 0.005917159760 + 0.1538461538x + x2 = 4.23668639076 Factor a perfect square on the left side: (x + 0.0769230769)(x + 0.0769230769) = 4.23668639076 Calculate the square root of the right side: 2.058321255 Break this problem into two subproblems by setting (x + 0.0769230769) equal to 2.058321255 and -2.058321255.Subproblem 1
x + 0.0769230769 = 2.058321255 Simplifying x + 0.0769230769 = 2.058321255 Reorder the terms: 0.0769230769 + x = 2.058321255 Solving 0.0769230769 + x = 2.058321255 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0769230769' to each side of the equation. 0.0769230769 + -0.0769230769 + x = 2.058321255 + -0.0769230769 Combine like terms: 0.0769230769 + -0.0769230769 = 0.0000000000 0.0000000000 + x = 2.058321255 + -0.0769230769 x = 2.058321255 + -0.0769230769 Combine like terms: 2.058321255 + -0.0769230769 = 1.9813981781 x = 1.9813981781 Simplifying x = 1.9813981781Subproblem 2
x + 0.0769230769 = -2.058321255 Simplifying x + 0.0769230769 = -2.058321255 Reorder the terms: 0.0769230769 + x = -2.058321255 Solving 0.0769230769 + x = -2.058321255 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0769230769' to each side of the equation. 0.0769230769 + -0.0769230769 + x = -2.058321255 + -0.0769230769 Combine like terms: 0.0769230769 + -0.0769230769 = 0.0000000000 0.0000000000 + x = -2.058321255 + -0.0769230769 x = -2.058321255 + -0.0769230769 Combine like terms: -2.058321255 + -0.0769230769 = -2.1352443319 x = -2.1352443319 Simplifying x = -2.1352443319Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.9813981781, -2.1352443319}
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