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Simplifying 2x2 + -6 + 2x + x2 + 12 = 78 Reorder the terms: -6 + 12 + 2x + 2x2 + x2 = 78 Combine like terms: -6 + 12 = 6 6 + 2x + 2x2 + x2 = 78 Combine like terms: 2x2 + x2 = 3x2 6 + 2x + 3x2 = 78 Solving 6 + 2x + 3x2 = 78 Solving for variable 'x'. Reorder the terms: 6 + -78 + 2x + 3x2 = 78 + -78 Combine like terms: 6 + -78 = -72 -72 + 2x + 3x2 = 78 + -78 Combine like terms: 78 + -78 = 0 -72 + 2x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -24 + 0.6666666667x + x2 = 0 Move the constant term to the right: Add '24' to each side of the equation. -24 + 0.6666666667x + 24 + x2 = 0 + 24 Reorder the terms: -24 + 24 + 0.6666666667x + x2 = 0 + 24 Combine like terms: -24 + 24 = 0 0 + 0.6666666667x + x2 = 0 + 24 0.6666666667x + x2 = 0 + 24 Combine like terms: 0 + 24 = 24 0.6666666667x + x2 = 24 The x term is 0.6666666667x. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667x + 0.1111111112 + x2 = 24 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667x + x2 = 24 + 0.1111111112 Combine like terms: 24 + 0.1111111112 = 24.1111111112 0.1111111112 + 0.6666666667x + x2 = 24.1111111112 Factor a perfect square on the left side: (x + 0.3333333334)(x + 0.3333333334) = 24.1111111112 Calculate the square root of the right side: 4.910306621 Break this problem into two subproblems by setting (x + 0.3333333334) equal to 4.910306621 and -4.910306621.Subproblem 1
x + 0.3333333334 = 4.910306621 Simplifying x + 0.3333333334 = 4.910306621 Reorder the terms: 0.3333333334 + x = 4.910306621 Solving 0.3333333334 + x = 4.910306621 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = 4.910306621 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 4.910306621 + -0.3333333334 x = 4.910306621 + -0.3333333334 Combine like terms: 4.910306621 + -0.3333333334 = 4.5769732876 x = 4.5769732876 Simplifying x = 4.5769732876Subproblem 2
x + 0.3333333334 = -4.910306621 Simplifying x + 0.3333333334 = -4.910306621 Reorder the terms: 0.3333333334 + x = -4.910306621 Solving 0.3333333334 + x = -4.910306621 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = -4.910306621 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -4.910306621 + -0.3333333334 x = -4.910306621 + -0.3333333334 Combine like terms: -4.910306621 + -0.3333333334 = -5.2436399544 x = -5.2436399544 Simplifying x = -5.2436399544Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.5769732876, -5.2436399544}
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