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Simplifying (5y + -3)(2y + 1) = 3 Reorder the terms: (-3 + 5y)(2y + 1) = 3 Reorder the terms: (-3 + 5y)(1 + 2y) = 3 Multiply (-3 + 5y) * (1 + 2y) (-3(1 + 2y) + 5y * (1 + 2y)) = 3 ((1 * -3 + 2y * -3) + 5y * (1 + 2y)) = 3 ((-3 + -6y) + 5y * (1 + 2y)) = 3 (-3 + -6y + (1 * 5y + 2y * 5y)) = 3 (-3 + -6y + (5y + 10y2)) = 3 Combine like terms: -6y + 5y = -1y (-3 + -1y + 10y2) = 3 Solving -3 + -1y + 10y2 = 3 Solving for variable 'y'. Reorder the terms: -3 + -3 + -1y + 10y2 = 3 + -3 Combine like terms: -3 + -3 = -6 -6 + -1y + 10y2 = 3 + -3 Combine like terms: 3 + -3 = 0 -6 + -1y + 10y2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.6 + -0.1y + y2 = 0.0 Move the constant term to the right: Add '0.6' to each side of the equation. -0.6 + -0.1y + 0.6 + y2 = 0.0 + 0.6 Reorder the terms: -0.6 + 0.6 + -0.1y + y2 = 0.0 + 0.6 Combine like terms: -0.6 + 0.6 = 0.0 0.0 + -0.1y + y2 = 0.0 + 0.6 -0.1y + y2 = 0.0 + 0.6 Combine like terms: 0.0 + 0.6 = 0.6 -0.1y + y2 = 0.6 The y term is -0.1y. Take half its coefficient (-0.05). Square it (0.0025) and add it to both sides. Add '0.0025' to each side of the equation. -0.1y + 0.0025 + y2 = 0.6 + 0.0025 Reorder the terms: 0.0025 + -0.1y + y2 = 0.6 + 0.0025 Combine like terms: 0.6 + 0.0025 = 0.6025 0.0025 + -0.1y + y2 = 0.6025 Factor a perfect square on the left side: (y + -0.05)(y + -0.05) = 0.6025 Calculate the square root of the right side: 0.776208735 Break this problem into two subproblems by setting (y + -0.05) equal to 0.776208735 and -0.776208735.Subproblem 1
y + -0.05 = 0.776208735 Simplifying y + -0.05 = 0.776208735 Reorder the terms: -0.05 + y = 0.776208735 Solving -0.05 + y = 0.776208735 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.05' to each side of the equation. -0.05 + 0.05 + y = 0.776208735 + 0.05 Combine like terms: -0.05 + 0.05 = 0.00 0.00 + y = 0.776208735 + 0.05 y = 0.776208735 + 0.05 Combine like terms: 0.776208735 + 0.05 = 0.826208735 y = 0.826208735 Simplifying y = 0.826208735Subproblem 2
y + -0.05 = -0.776208735 Simplifying y + -0.05 = -0.776208735 Reorder the terms: -0.05 + y = -0.776208735 Solving -0.05 + y = -0.776208735 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.05' to each side of the equation. -0.05 + 0.05 + y = -0.776208735 + 0.05 Combine like terms: -0.05 + 0.05 = 0.00 0.00 + y = -0.776208735 + 0.05 y = -0.776208735 + 0.05 Combine like terms: -0.776208735 + 0.05 = -0.726208735 y = -0.726208735 Simplifying y = -0.726208735Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.826208735, -0.726208735}
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