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Simplifying -3 = -4(-7y + 1) * 5(8 + 2y) Reorder the terms: -3 = -4(1 + -7y) * 5(8 + 2y) Reorder the terms for easier multiplication: -3 = -4 * 5(1 + -7y)(8 + 2y) Multiply -4 * 5 -3 = -20(1 + -7y)(8 + 2y) Multiply (1 + -7y) * (8 + 2y) -3 = -20(1(8 + 2y) + -7y * (8 + 2y)) -3 = -20((8 * 1 + 2y * 1) + -7y * (8 + 2y)) -3 = -20((8 + 2y) + -7y * (8 + 2y)) -3 = -20(8 + 2y + (8 * -7y + 2y * -7y)) -3 = -20(8 + 2y + (-56y + -14y2)) Combine like terms: 2y + -56y = -54y -3 = -20(8 + -54y + -14y2) -3 = (8 * -20 + -54y * -20 + -14y2 * -20) -3 = (-160 + 1080y + 280y2) Solving -3 = -160 + 1080y + 280y2 Solving for variable 'y'. Combine like terms: -3 + 160 = 157 157 + -1080y + -280y2 = -160 + 1080y + 280y2 + 160 + -1080y + -280y2 Reorder the terms: 157 + -1080y + -280y2 = -160 + 160 + 1080y + -1080y + 280y2 + -280y2 Combine like terms: -160 + 160 = 0 157 + -1080y + -280y2 = 0 + 1080y + -1080y + 280y2 + -280y2 157 + -1080y + -280y2 = 1080y + -1080y + 280y2 + -280y2 Combine like terms: 1080y + -1080y = 0 157 + -1080y + -280y2 = 0 + 280y2 + -280y2 157 + -1080y + -280y2 = 280y2 + -280y2 Combine like terms: 280y2 + -280y2 = 0 157 + -1080y + -280y2 = 0 Begin completing the square. Divide all terms by -280 the coefficient of the squared term: Divide each side by '-280'. -0.5607142857 + 3.857142857y + y2 = 0 Move the constant term to the right: Add '0.5607142857' to each side of the equation. -0.5607142857 + 3.857142857y + 0.5607142857 + y2 = 0 + 0.5607142857 Reorder the terms: -0.5607142857 + 0.5607142857 + 3.857142857y + y2 = 0 + 0.5607142857 Combine like terms: -0.5607142857 + 0.5607142857 = 0.0000000000 0.0000000000 + 3.857142857y + y2 = 0 + 0.5607142857 3.857142857y + y2 = 0 + 0.5607142857 Combine like terms: 0 + 0.5607142857 = 0.5607142857 3.857142857y + y2 = 0.5607142857 The y term is 3.857142857y. Take half its coefficient (1.928571429). Square it (3.719387757) and add it to both sides. Add '3.719387757' to each side of the equation. 3.857142857y + 3.719387757 + y2 = 0.5607142857 + 3.719387757 Reorder the terms: 3.719387757 + 3.857142857y + y2 = 0.5607142857 + 3.719387757 Combine like terms: 0.5607142857 + 3.719387757 = 4.2801020427 3.719387757 + 3.857142857y + y2 = 4.2801020427 Factor a perfect square on the left side: (y + 1.928571429)(y + 1.928571429) = 4.2801020427 Calculate the square root of the right side: 2.068840749 Break this problem into two subproblems by setting (y + 1.928571429) equal to 2.068840749 and -2.068840749.Subproblem 1
y + 1.928571429 = 2.068840749 Simplifying y + 1.928571429 = 2.068840749 Reorder the terms: 1.928571429 + y = 2.068840749 Solving 1.928571429 + y = 2.068840749 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.928571429' to each side of the equation. 1.928571429 + -1.928571429 + y = 2.068840749 + -1.928571429 Combine like terms: 1.928571429 + -1.928571429 = 0.000000000 0.000000000 + y = 2.068840749 + -1.928571429 y = 2.068840749 + -1.928571429 Combine like terms: 2.068840749 + -1.928571429 = 0.14026932 y = 0.14026932 Simplifying y = 0.14026932Subproblem 2
y + 1.928571429 = -2.068840749 Simplifying y + 1.928571429 = -2.068840749 Reorder the terms: 1.928571429 + y = -2.068840749 Solving 1.928571429 + y = -2.068840749 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.928571429' to each side of the equation. 1.928571429 + -1.928571429 + y = -2.068840749 + -1.928571429 Combine like terms: 1.928571429 + -1.928571429 = 0.000000000 0.000000000 + y = -2.068840749 + -1.928571429 y = -2.068840749 + -1.928571429 Combine like terms: -2.068840749 + -1.928571429 = -3.997412178 y = -3.997412178 Simplifying y = -3.997412178Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.14026932, -3.997412178}
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