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Simplifying 42x2 = 27x + 40 Reorder the terms: 42x2 = 40 + 27x Solving 42x2 = 40 + 27x Solving for variable 'x'. Reorder the terms: -40 + -27x + 42x2 = 40 + 27x + -40 + -27x Reorder the terms: -40 + -27x + 42x2 = 40 + -40 + 27x + -27x Combine like terms: 40 + -40 = 0 -40 + -27x + 42x2 = 0 + 27x + -27x -40 + -27x + 42x2 = 27x + -27x Combine like terms: 27x + -27x = 0 -40 + -27x + 42x2 = 0 Begin completing the square. Divide all terms by 42 the coefficient of the squared term: Divide each side by '42'. -0.9523809524 + -0.6428571429x + x2 = 0 Move the constant term to the right: Add '0.9523809524' to each side of the equation. -0.9523809524 + -0.6428571429x + 0.9523809524 + x2 = 0 + 0.9523809524 Reorder the terms: -0.9523809524 + 0.9523809524 + -0.6428571429x + x2 = 0 + 0.9523809524 Combine like terms: -0.9523809524 + 0.9523809524 = 0.0000000000 0.0000000000 + -0.6428571429x + x2 = 0 + 0.9523809524 -0.6428571429x + x2 = 0 + 0.9523809524 Combine like terms: 0 + 0.9523809524 = 0.9523809524 -0.6428571429x + x2 = 0.9523809524 The x term is -0.6428571429x. Take half its coefficient (-0.3214285715). Square it (0.1033163266) and add it to both sides. Add '0.1033163266' to each side of the equation. -0.6428571429x + 0.1033163266 + x2 = 0.9523809524 + 0.1033163266 Reorder the terms: 0.1033163266 + -0.6428571429x + x2 = 0.9523809524 + 0.1033163266 Combine like terms: 0.9523809524 + 0.1033163266 = 1.055697279 0.1033163266 + -0.6428571429x + x2 = 1.055697279 Factor a perfect square on the left side: (x + -0.3214285715)(x + -0.3214285715) = 1.055697279 Calculate the square root of the right side: 1.027471303 Break this problem into two subproblems by setting (x + -0.3214285715) equal to 1.027471303 and -1.027471303.Subproblem 1
x + -0.3214285715 = 1.027471303 Simplifying x + -0.3214285715 = 1.027471303 Reorder the terms: -0.3214285715 + x = 1.027471303 Solving -0.3214285715 + x = 1.027471303 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3214285715' to each side of the equation. -0.3214285715 + 0.3214285715 + x = 1.027471303 + 0.3214285715 Combine like terms: -0.3214285715 + 0.3214285715 = 0.0000000000 0.0000000000 + x = 1.027471303 + 0.3214285715 x = 1.027471303 + 0.3214285715 Combine like terms: 1.027471303 + 0.3214285715 = 1.3488998745 x = 1.3488998745 Simplifying x = 1.3488998745Subproblem 2
x + -0.3214285715 = -1.027471303 Simplifying x + -0.3214285715 = -1.027471303 Reorder the terms: -0.3214285715 + x = -1.027471303 Solving -0.3214285715 + x = -1.027471303 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3214285715' to each side of the equation. -0.3214285715 + 0.3214285715 + x = -1.027471303 + 0.3214285715 Combine like terms: -0.3214285715 + 0.3214285715 = 0.0000000000 0.0000000000 + x = -1.027471303 + 0.3214285715 x = -1.027471303 + 0.3214285715 Combine like terms: -1.027471303 + 0.3214285715 = -0.7060427315 x = -0.7060427315 Simplifying x = -0.7060427315Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.3488998745, -0.7060427315}
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