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Simplifying 4x2 + 7x = x2 + -1x + 4 Reorder the terms: 7x + 4x2 = x2 + -1x + 4 Reorder the terms: 7x + 4x2 = 4 + -1x + x2 Solving 7x + 4x2 = 4 + -1x + x2 Solving for variable 'x'. Reorder the terms: -4 + 7x + x + 4x2 + -1x2 = 4 + -1x + x2 + -4 + x + -1x2 Combine like terms: 7x + x = 8x -4 + 8x + 4x2 + -1x2 = 4 + -1x + x2 + -4 + x + -1x2 Combine like terms: 4x2 + -1x2 = 3x2 -4 + 8x + 3x2 = 4 + -1x + x2 + -4 + x + -1x2 Reorder the terms: -4 + 8x + 3x2 = 4 + -4 + -1x + x + x2 + -1x2 Combine like terms: 4 + -4 = 0 -4 + 8x + 3x2 = 0 + -1x + x + x2 + -1x2 -4 + 8x + 3x2 = -1x + x + x2 + -1x2 Combine like terms: -1x + x = 0 -4 + 8x + 3x2 = 0 + x2 + -1x2 -4 + 8x + 3x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -4 + 8x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1.333333333 + 2.666666667x + x2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + 2.666666667x + 1.333333333 + x2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + 2.666666667x + x2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + 2.666666667x + x2 = 0 + 1.333333333 2.666666667x + x2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 2.666666667x + x2 = 1.333333333 The x term is 2.666666667x. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667x + 1.777777780 + x2 = 1.333333333 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667x + x2 = 1.333333333 + 1.777777780 Combine like terms: 1.333333333 + 1.777777780 = 3.111111113 1.777777780 + 2.666666667x + x2 = 3.111111113 Factor a perfect square on the left side: (x + 1.333333334)(x + 1.333333334) = 3.111111113 Calculate the square root of the right side: 1.763834208 Break this problem into two subproblems by setting (x + 1.333333334) equal to 1.763834208 and -1.763834208.Subproblem 1
x + 1.333333334 = 1.763834208 Simplifying x + 1.333333334 = 1.763834208 Reorder the terms: 1.333333334 + x = 1.763834208 Solving 1.333333334 + x = 1.763834208 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = 1.763834208 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 1.763834208 + -1.333333334 x = 1.763834208 + -1.333333334 Combine like terms: 1.763834208 + -1.333333334 = 0.430500874 x = 0.430500874 Simplifying x = 0.430500874Subproblem 2
x + 1.333333334 = -1.763834208 Simplifying x + 1.333333334 = -1.763834208 Reorder the terms: 1.333333334 + x = -1.763834208 Solving 1.333333334 + x = -1.763834208 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = -1.763834208 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -1.763834208 + -1.333333334 x = -1.763834208 + -1.333333334 Combine like terms: -1.763834208 + -1.333333334 = -3.097167542 x = -3.097167542 Simplifying x = -3.097167542Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.430500874, -3.097167542}
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