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Simplifying 5y(3y + -2) + (2y + -3) = 0 Reorder the terms: 5y(-2 + 3y) + (2y + -3) = 0 (-2 * 5y + 3y * 5y) + (2y + -3) = 0 (-10y + 15y2) + (2y + -3) = 0 Reorder the terms: -10y + 15y2 + (-3 + 2y) = 0 Remove parenthesis around (-3 + 2y) -10y + 15y2 + -3 + 2y = 0 Reorder the terms: -3 + -10y + 2y + 15y2 = 0 Combine like terms: -10y + 2y = -8y -3 + -8y + 15y2 = 0 Solving -3 + -8y + 15y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -0.2 + -0.5333333333y + y2 = 0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + -0.5333333333y + 0.2 + y2 = 0 + 0.2 Reorder the terms: -0.2 + 0.2 + -0.5333333333y + y2 = 0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + -0.5333333333y + y2 = 0 + 0.2 -0.5333333333y + y2 = 0 + 0.2 Combine like terms: 0 + 0.2 = 0.2 -0.5333333333y + y2 = 0.2 The y term is -0.5333333333y. Take half its coefficient (-0.2666666667). Square it (0.07111111113) and add it to both sides. Add '0.07111111113' to each side of the equation. -0.5333333333y + 0.07111111113 + y2 = 0.2 + 0.07111111113 Reorder the terms: 0.07111111113 + -0.5333333333y + y2 = 0.2 + 0.07111111113 Combine like terms: 0.2 + 0.07111111113 = 0.27111111113 0.07111111113 + -0.5333333333y + y2 = 0.27111111113 Factor a perfect square on the left side: (y + -0.2666666667)(y + -0.2666666667) = 0.27111111113 Calculate the square root of the right side: 0.520683312 Break this problem into two subproblems by setting (y + -0.2666666667) equal to 0.520683312 and -0.520683312.Subproblem 1
y + -0.2666666667 = 0.520683312 Simplifying y + -0.2666666667 = 0.520683312 Reorder the terms: -0.2666666667 + y = 0.520683312 Solving -0.2666666667 + y = 0.520683312 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.2666666667' to each side of the equation. -0.2666666667 + 0.2666666667 + y = 0.520683312 + 0.2666666667 Combine like terms: -0.2666666667 + 0.2666666667 = 0.0000000000 0.0000000000 + y = 0.520683312 + 0.2666666667 y = 0.520683312 + 0.2666666667 Combine like terms: 0.520683312 + 0.2666666667 = 0.7873499787 y = 0.7873499787 Simplifying y = 0.7873499787Subproblem 2
y + -0.2666666667 = -0.520683312 Simplifying y + -0.2666666667 = -0.520683312 Reorder the terms: -0.2666666667 + y = -0.520683312 Solving -0.2666666667 + y = -0.520683312 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.2666666667' to each side of the equation. -0.2666666667 + 0.2666666667 + y = -0.520683312 + 0.2666666667 Combine like terms: -0.2666666667 + 0.2666666667 = 0.0000000000 0.0000000000 + y = -0.520683312 + 0.2666666667 y = -0.520683312 + 0.2666666667 Combine like terms: -0.520683312 + 0.2666666667 = -0.2540166453 y = -0.2540166453 Simplifying y = -0.2540166453Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.7873499787, -0.2540166453}
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