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Simplifying 6x2 + -4x + -3 = 105 Reorder the terms: -3 + -4x + 6x2 = 105 Solving -3 + -4x + 6x2 = 105 Solving for variable 'x'. Reorder the terms: -3 + -105 + -4x + 6x2 = 105 + -105 Combine like terms: -3 + -105 = -108 -108 + -4x + 6x2 = 105 + -105 Combine like terms: 105 + -105 = 0 -108 + -4x + 6x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-54 + -2x + 3x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-54 + -2x + 3x2)' equal to zero and attempt to solve: Simplifying -54 + -2x + 3x2 = 0 Solving -54 + -2x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -18 + -0.6666666667x + x2 = 0 Move the constant term to the right: Add '18' to each side of the equation. -18 + -0.6666666667x + 18 + x2 = 0 + 18 Reorder the terms: -18 + 18 + -0.6666666667x + x2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + -0.6666666667x + x2 = 0 + 18 -0.6666666667x + x2 = 0 + 18 Combine like terms: 0 + 18 = 18 -0.6666666667x + x2 = 18 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 = 18 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667x + x2 = 18 + 0.1111111112 Combine like terms: 18 + 0.1111111112 = 18.1111111112 0.1111111112 + -0.6666666667x + x2 = 18.1111111112 Factor a perfect square on the left side: (x + -0.3333333334)(x + -0.3333333334) = 18.1111111112 Calculate the square root of the right side: 4.255715112 Break this problem into two subproblems by setting (x + -0.3333333334) equal to 4.255715112 and -4.255715112.Subproblem 1
x + -0.3333333334 = 4.255715112 Simplifying x + -0.3333333334 = 4.255715112 Reorder the terms: -0.3333333334 + x = 4.255715112 Solving -0.3333333334 + x = 4.255715112 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 = 4.255715112 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = 4.255715112 + 0.3333333334 x = 4.255715112 + 0.3333333334 Combine like terms: 4.255715112 + 0.3333333334 = 4.5890484454 x = 4.5890484454 Simplifying x = 4.5890484454Subproblem 2
x + -0.3333333334 = -4.255715112 Simplifying x + -0.3333333334 = -4.255715112 Reorder the terms: -0.3333333334 + x = -4.255715112 Solving -0.3333333334 + x = -4.255715112 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 = -4.255715112 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = -4.255715112 + 0.3333333334 x = -4.255715112 + 0.3333333334 Combine like terms: -4.255715112 + 0.3333333334 = -3.9223817786 x = -3.9223817786 Simplifying x = -3.9223817786Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.5890484454, -3.9223817786}Solution
x = {4.5890484454, -3.9223817786}
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