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Simplifying 6x2 + 78x + 175 = 0 Reorder the terms: 175 + 78x + 6x2 = 0 Solving 175 + 78x + 6x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 29.16666667 + 13x + x2 = 0 Move the constant term to the right: Add '-29.16666667' to each side of the equation. 29.16666667 + 13x + -29.16666667 + x2 = 0 + -29.16666667 Reorder the terms: 29.16666667 + -29.16666667 + 13x + x2 = 0 + -29.16666667 Combine like terms: 29.16666667 + -29.16666667 = 0.00000000 0.00000000 + 13x + x2 = 0 + -29.16666667 13x + x2 = 0 + -29.16666667 Combine like terms: 0 + -29.16666667 = -29.16666667 13x + x2 = -29.16666667 The x term is 13x. Take half its coefficient (6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. 13x + 42.25 + x2 = -29.16666667 + 42.25 Reorder the terms: 42.25 + 13x + x2 = -29.16666667 + 42.25 Combine like terms: -29.16666667 + 42.25 = 13.08333333 42.25 + 13x + x2 = 13.08333333 Factor a perfect square on the left side: (x + 6.5)(x + 6.5) = 13.08333333 Calculate the square root of the right side: 3.617089069 Break this problem into two subproblems by setting (x + 6.5) equal to 3.617089069 and -3.617089069.Subproblem 1
x + 6.5 = 3.617089069 Simplifying x + 6.5 = 3.617089069 Reorder the terms: 6.5 + x = 3.617089069 Solving 6.5 + x = 3.617089069 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + x = 3.617089069 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + x = 3.617089069 + -6.5 x = 3.617089069 + -6.5 Combine like terms: 3.617089069 + -6.5 = -2.882910931 x = -2.882910931 Simplifying x = -2.882910931Subproblem 2
x + 6.5 = -3.617089069 Simplifying x + 6.5 = -3.617089069 Reorder the terms: 6.5 + x = -3.617089069 Solving 6.5 + x = -3.617089069 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + x = -3.617089069 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + x = -3.617089069 + -6.5 x = -3.617089069 + -6.5 Combine like terms: -3.617089069 + -6.5 = -10.117089069 x = -10.117089069 Simplifying x = -10.117089069Solution
The solution to the problem is based on the solutions from the subproblems. x = {-2.882910931, -10.117089069}
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