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Simplifying (y2 + y) + -7(4y2 + -5y) + 2 = 0 Reorder the terms: (y + y2) + -7(4y2 + -5y) + 2 = 0 Remove parenthesis around (y + y2) y + y2 + -7(4y2 + -5y) + 2 = 0 Reorder the terms: y + y2 + -7(-5y + 4y2) + 2 = 0 y + y2 + (-5y * -7 + 4y2 * -7) + 2 = 0 y + y2 + (35y + -28y2) + 2 = 0 Reorder the terms: 2 + y + 35y + y2 + -28y2 = 0 Combine like terms: y + 35y = 36y 2 + 36y + y2 + -28y2 = 0 Combine like terms: y2 + -28y2 = -27y2 2 + 36y + -27y2 = 0 Solving 2 + 36y + -27y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by -27 the coefficient of the squared term: Divide each side by '-27'. -0.07407407407 + -1.333333333y + y2 = 0 Move the constant term to the right: Add '0.07407407407' to each side of the equation. -0.07407407407 + -1.333333333y + 0.07407407407 + y2 = 0 + 0.07407407407 Reorder the terms: -0.07407407407 + 0.07407407407 + -1.333333333y + y2 = 0 + 0.07407407407 Combine like terms: -0.07407407407 + 0.07407407407 = 0.00000000000 0.00000000000 + -1.333333333y + y2 = 0 + 0.07407407407 -1.333333333y + y2 = 0 + 0.07407407407 Combine like terms: 0 + 0.07407407407 = 0.07407407407 -1.333333333y + y2 = 0.07407407407 The y term is -1.333333333y. Take half its coefficient (-0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. -1.333333333y + 0.4444444442 + y2 = 0.07407407407 + 0.4444444442 Reorder the terms: 0.4444444442 + -1.333333333y + y2 = 0.07407407407 + 0.4444444442 Combine like terms: 0.07407407407 + 0.4444444442 = 0.51851851827 0.4444444442 + -1.333333333y + y2 = 0.51851851827 Factor a perfect square on the left side: (y + -0.6666666665)(y + -0.6666666665) = 0.51851851827 Calculate the square root of the right side: 0.7200823 Break this problem into two subproblems by setting (y + -0.6666666665) equal to 0.7200823 and -0.7200823.Subproblem 1
y + -0.6666666665 = 0.7200823 Simplifying y + -0.6666666665 = 0.7200823 Reorder the terms: -0.6666666665 + y = 0.7200823 Solving -0.6666666665 + y = 0.7200823 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + y = 0.7200823 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + y = 0.7200823 + 0.6666666665 y = 0.7200823 + 0.6666666665 Combine like terms: 0.7200823 + 0.6666666665 = 1.3867489665 y = 1.3867489665 Simplifying y = 1.3867489665Subproblem 2
y + -0.6666666665 = -0.7200823 Simplifying y + -0.6666666665 = -0.7200823 Reorder the terms: -0.6666666665 + y = -0.7200823 Solving -0.6666666665 + y = -0.7200823 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + y = -0.7200823 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + y = -0.7200823 + 0.6666666665 y = -0.7200823 + 0.6666666665 Combine like terms: -0.7200823 + 0.6666666665 = -0.0534156335 y = -0.0534156335 Simplifying y = -0.0534156335Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.3867489665, -0.0534156335}
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