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Simplifying (2x2 + -5x + 12) + (6x2 + 8x + -78) = 0 Reorder the terms: (12 + -5x + 2x2) + (6x2 + 8x + -78) = 0 Remove parenthesis around (12 + -5x + 2x2) 12 + -5x + 2x2 + (6x2 + 8x + -78) = 0 Reorder the terms: 12 + -5x + 2x2 + (-78 + 8x + 6x2) = 0 Remove parenthesis around (-78 + 8x + 6x2) 12 + -5x + 2x2 + -78 + 8x + 6x2 = 0 Reorder the terms: 12 + -78 + -5x + 8x + 2x2 + 6x2 = 0 Combine like terms: 12 + -78 = -66 -66 + -5x + 8x + 2x2 + 6x2 = 0 Combine like terms: -5x + 8x = 3x -66 + 3x + 2x2 + 6x2 = 0 Combine like terms: 2x2 + 6x2 = 8x2 -66 + 3x + 8x2 = 0 Solving -66 + 3x + 8x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -8.25 + 0.375x + x2 = 0 Move the constant term to the right: Add '8.25' to each side of the equation. -8.25 + 0.375x + 8.25 + x2 = 0 + 8.25 Reorder the terms: -8.25 + 8.25 + 0.375x + x2 = 0 + 8.25 Combine like terms: -8.25 + 8.25 = 0.00 0.00 + 0.375x + x2 = 0 + 8.25 0.375x + x2 = 0 + 8.25 Combine like terms: 0 + 8.25 = 8.25 0.375x + x2 = 8.25 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 = 8.25 + 0.03515625 Reorder the terms: 0.03515625 + 0.375x + x2 = 8.25 + 0.03515625 Combine like terms: 8.25 + 0.03515625 = 8.28515625 0.03515625 + 0.375x + x2 = 8.28515625 Factor a perfect square on the left side: (x + 0.1875)(x + 0.1875) = 8.28515625 Calculate the square root of the right side: 2.878394735 Break this problem into two subproblems by setting (x + 0.1875) equal to 2.878394735 and -2.878394735.Subproblem 1
x + 0.1875 = 2.878394735 Simplifying x + 0.1875 = 2.878394735 Reorder the terms: 0.1875 + x = 2.878394735 Solving 0.1875 + x = 2.878394735 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 = 2.878394735 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + x = 2.878394735 + -0.1875 x = 2.878394735 + -0.1875 Combine like terms: 2.878394735 + -0.1875 = 2.690894735 x = 2.690894735 Simplifying x = 2.690894735Subproblem 2
x + 0.1875 = -2.878394735 Simplifying x + 0.1875 = -2.878394735 Reorder the terms: 0.1875 + x = -2.878394735 Solving 0.1875 + x = -2.878394735 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 = -2.878394735 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + x = -2.878394735 + -0.1875 x = -2.878394735 + -0.1875 Combine like terms: -2.878394735 + -0.1875 = -3.065894735 x = -3.065894735 Simplifying x = -3.065894735Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.690894735, -3.065894735}
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