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Simplifying 12x2 + 67x + 66 = 0 Reorder the terms: 66 + 67x + 12x2 = 0 Solving 66 + 67x + 12x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 5.5 + 5.583333333x + x2 = 0 Move the constant term to the right: Add '-5.5' to each side of the equation. 5.5 + 5.583333333x + -5.5 + x2 = 0 + -5.5 Reorder the terms: 5.5 + -5.5 + 5.583333333x + x2 = 0 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + 5.583333333x + x2 = 0 + -5.5 5.583333333x + x2 = 0 + -5.5 Combine like terms: 0 + -5.5 = -5.5 5.583333333x + x2 = -5.5 The x term is 5.583333333x. Take half its coefficient (2.791666667). Square it (7.793402780) and add it to both sides. Add '7.793402780' to each side of the equation. 5.583333333x + 7.793402780 + x2 = -5.5 + 7.793402780 Reorder the terms: 7.793402780 + 5.583333333x + x2 = -5.5 + 7.793402780 Combine like terms: -5.5 + 7.793402780 = 2.29340278 7.793402780 + 5.583333333x + x2 = 2.29340278 Factor a perfect square on the left side: (x + 2.791666667)(x + 2.791666667) = 2.29340278 Calculate the square root of the right side: 1.514398488 Break this problem into two subproblems by setting (x + 2.791666667) equal to 1.514398488 and -1.514398488.Subproblem 1
x + 2.791666667 = 1.514398488 Simplifying x + 2.791666667 = 1.514398488 Reorder the terms: 2.791666667 + x = 1.514398488 Solving 2.791666667 + x = 1.514398488 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.791666667' to each side of the equation. 2.791666667 + -2.791666667 + x = 1.514398488 + -2.791666667 Combine like terms: 2.791666667 + -2.791666667 = 0.000000000 0.000000000 + x = 1.514398488 + -2.791666667 x = 1.514398488 + -2.791666667 Combine like terms: 1.514398488 + -2.791666667 = -1.277268179 x = -1.277268179 Simplifying x = -1.277268179Subproblem 2
x + 2.791666667 = -1.514398488 Simplifying x + 2.791666667 = -1.514398488 Reorder the terms: 2.791666667 + x = -1.514398488 Solving 2.791666667 + x = -1.514398488 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.791666667' to each side of the equation. 2.791666667 + -2.791666667 + x = -1.514398488 + -2.791666667 Combine like terms: 2.791666667 + -2.791666667 = 0.000000000 0.000000000 + x = -1.514398488 + -2.791666667 x = -1.514398488 + -2.791666667 Combine like terms: -1.514398488 + -2.791666667 = -4.306065155 x = -4.306065155 Simplifying x = -4.306065155Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.277268179, -4.306065155}
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