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Simplifying (5y + -2)[y + -3(y + -5)] = 6 Reorder the terms: (-2 + 5y)[y + -3(y + -5)] = 6 Reorder the terms: (-2 + 5y)[y + -3(-5 + y)] = 6 (-2 + 5y)[y + (-5 * -3 + y * -3)] = 6 (-2 + 5y)[y + (15 + -3y)] = 6 Reorder the terms: (-2 + 5y)[15 + y + -3y] = 6 Combine like terms: y + -3y = -2y (-2 + 5y)[15 + -2y] = 6 Multiply (-2 + 5y) * [15 + -2y] (-2[15 + -2y] + 5y * [15 + -2y]) = 6 ([15 * -2 + -2y * -2] + 5y * [15 + -2y]) = 6 ([-30 + 4y] + 5y * [15 + -2y]) = 6 (-30 + 4y + [15 * 5y + -2y * 5y]) = 6 (-30 + 4y + [75y + -10y2]) = 6 Combine like terms: 4y + 75y = 79y (-30 + 79y + -10y2) = 6 Solving -30 + 79y + -10y2 = 6 Solving for variable 'y'. Reorder the terms: -30 + -6 + 79y + -10y2 = 6 + -6 Combine like terms: -30 + -6 = -36 -36 + 79y + -10y2 = 6 + -6 Combine like terms: 6 + -6 = 0 -36 + 79y + -10y2 = 0 Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. 3.6 + -7.9y + y2 = 0.0 Move the constant term to the right: Add '-3.6' to each side of the equation. 3.6 + -7.9y + -3.6 + y2 = 0.0 + -3.6 Reorder the terms: 3.6 + -3.6 + -7.9y + y2 = 0.0 + -3.6 Combine like terms: 3.6 + -3.6 = 0.0 0.0 + -7.9y + y2 = 0.0 + -3.6 -7.9y + y2 = 0.0 + -3.6 Combine like terms: 0.0 + -3.6 = -3.6 -7.9y + y2 = -3.6 The y term is -7.9y. Take half its coefficient (-3.95). Square it (15.6025) and add it to both sides. Add '15.6025' to each side of the equation. -7.9y + 15.6025 + y2 = -3.6 + 15.6025 Reorder the terms: 15.6025 + -7.9y + y2 = -3.6 + 15.6025 Combine like terms: -3.6 + 15.6025 = 12.0025 15.6025 + -7.9y + y2 = 12.0025 Factor a perfect square on the left side: (y + -3.95)(y + -3.95) = 12.0025 Calculate the square root of the right side: 3.46446244 Break this problem into two subproblems by setting (y + -3.95) equal to 3.46446244 and -3.46446244.Subproblem 1
y + -3.95 = 3.46446244 Simplifying y + -3.95 = 3.46446244 Reorder the terms: -3.95 + y = 3.46446244 Solving -3.95 + y = 3.46446244 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.95' to each side of the equation. -3.95 + 3.95 + y = 3.46446244 + 3.95 Combine like terms: -3.95 + 3.95 = 0.00 0.00 + y = 3.46446244 + 3.95 y = 3.46446244 + 3.95 Combine like terms: 3.46446244 + 3.95 = 7.41446244 y = 7.41446244 Simplifying y = 7.41446244Subproblem 2
y + -3.95 = -3.46446244 Simplifying y + -3.95 = -3.46446244 Reorder the terms: -3.95 + y = -3.46446244 Solving -3.95 + y = -3.46446244 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.95' to each side of the equation. -3.95 + 3.95 + y = -3.46446244 + 3.95 Combine like terms: -3.95 + 3.95 = 0.00 0.00 + y = -3.46446244 + 3.95 y = -3.46446244 + 3.95 Combine like terms: -3.46446244 + 3.95 = 0.48553756 y = 0.48553756 Simplifying y = 0.48553756Solution
The solution to the problem is based on the solutions from the subproblems. y = {7.41446244, 0.48553756}
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