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Simplifying 4x + 3x(x + 1) = 85 Reorder the terms: 4x + 3x(1 + x) = 85 4x + (1 * 3x + x * 3x) = 85 4x + (3x + 3x2) = 85 Combine like terms: 4x + 3x = 7x 7x + 3x2 = 85 Solving 7x + 3x2 = 85 Solving for variable 'x'. Reorder the terms: -85 + 7x + 3x2 = 85 + -85 Combine like terms: 85 + -85 = 0 -85 + 7x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -28.33333333 + 2.333333333x + x2 = 0 Move the constant term to the right: Add '28.33333333' to each side of the equation. -28.33333333 + 2.333333333x + 28.33333333 + x2 = 0 + 28.33333333 Reorder the terms: -28.33333333 + 28.33333333 + 2.333333333x + x2 = 0 + 28.33333333 Combine like terms: -28.33333333 + 28.33333333 = 0.00000000 0.00000000 + 2.333333333x + x2 = 0 + 28.33333333 2.333333333x + x2 = 0 + 28.33333333 Combine like terms: 0 + 28.33333333 = 28.33333333 2.333333333x + x2 = 28.33333333 The x term is 2.333333333x. Take half its coefficient (1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. 2.333333333x + 1.361111112 + x2 = 28.33333333 + 1.361111112 Reorder the terms: 1.361111112 + 2.333333333x + x2 = 28.33333333 + 1.361111112 Combine like terms: 28.33333333 + 1.361111112 = 29.694444442 1.361111112 + 2.333333333x + x2 = 29.694444442 Factor a perfect square on the left side: (x + 1.166666667)(x + 1.166666667) = 29.694444442 Calculate the square root of the right side: 5.449260908 Break this problem into two subproblems by setting (x + 1.166666667) equal to 5.449260908 and -5.449260908.Subproblem 1
x + 1.166666667 = 5.449260908 Simplifying x + 1.166666667 = 5.449260908 Reorder the terms: 1.166666667 + x = 5.449260908 Solving 1.166666667 + x = 5.449260908 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + x = 5.449260908 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = 5.449260908 + -1.166666667 x = 5.449260908 + -1.166666667 Combine like terms: 5.449260908 + -1.166666667 = 4.282594241 x = 4.282594241 Simplifying x = 4.282594241Subproblem 2
x + 1.166666667 = -5.449260908 Simplifying x + 1.166666667 = -5.449260908 Reorder the terms: 1.166666667 + x = -5.449260908 Solving 1.166666667 + x = -5.449260908 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + x = -5.449260908 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = -5.449260908 + -1.166666667 x = -5.449260908 + -1.166666667 Combine like terms: -5.449260908 + -1.166666667 = -6.615927575 x = -6.615927575 Simplifying x = -6.615927575Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.282594241, -6.615927575}
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