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Simplifying 6x2 + 20x + 15 = 0 Reorder the terms: 15 + 20x + 6x2 = 0 Solving 15 + 20x + 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'. 2.5 + 3.333333333x + x2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + 3.333333333x + -2.5 + x2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + 3.333333333x + x2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + 3.333333333x + x2 = 0 + -2.5 3.333333333x + x2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 3.333333333x + x2 = -2.5 The x term is 3.333333333x. Take half its coefficient (1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. 3.333333333x + 2.777777779 + x2 = -2.5 + 2.777777779 Reorder the terms: 2.777777779 + 3.333333333x + x2 = -2.5 + 2.777777779 Combine like terms: -2.5 + 2.777777779 = 0.277777779 2.777777779 + 3.333333333x + x2 = 0.277777779 Factor a perfect square on the left side: (x + 1.666666667)(x + 1.666666667) = 0.277777779 Calculate the square root of the right side: 0.527046278 Break this problem into two subproblems by setting (x + 1.666666667) equal to 0.527046278 and -0.527046278.Subproblem 1
x + 1.666666667 = 0.527046278 Simplifying x + 1.666666667 = 0.527046278 Reorder the terms: 1.666666667 + x = 0.527046278 Solving 1.666666667 + x = 0.527046278 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + x = 0.527046278 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + x = 0.527046278 + -1.666666667 x = 0.527046278 + -1.666666667 Combine like terms: 0.527046278 + -1.666666667 = -1.139620389 x = -1.139620389 Simplifying x = -1.139620389Subproblem 2
x + 1.666666667 = -0.527046278 Simplifying x + 1.666666667 = -0.527046278 Reorder the terms: 1.666666667 + x = -0.527046278 Solving 1.666666667 + x = -0.527046278 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + x = -0.527046278 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + x = -0.527046278 + -1.666666667 x = -0.527046278 + -1.666666667 Combine like terms: -0.527046278 + -1.666666667 = -2.193712945 x = -2.193712945 Simplifying x = -2.193712945Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.139620389, -2.193712945}
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