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Simplifying 3y + 4 = 6y(4y + 6) Reorder the terms: 4 + 3y = 6y(4y + 6) Reorder the terms: 4 + 3y = 6y(6 + 4y) 4 + 3y = (6 * 6y + 4y * 6y) 4 + 3y = (36y + 24y2) Solving 4 + 3y = 36y + 24y2 Solving for variable 'y'. Combine like terms: 3y + -36y = -33y 4 + -33y + -24y2 = 36y + 24y2 + -36y + -24y2 Reorder the terms: 4 + -33y + -24y2 = 36y + -36y + 24y2 + -24y2 Combine like terms: 36y + -36y = 0 4 + -33y + -24y2 = 0 + 24y2 + -24y2 4 + -33y + -24y2 = 24y2 + -24y2 Combine like terms: 24y2 + -24y2 = 0 4 + -33y + -24y2 = 0 Begin completing the square. Divide all terms by -24 the coefficient of the squared term: Divide each side by '-24'. -0.1666666667 + 1.375y + y2 = 0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + 1.375y + 0.1666666667 + y2 = 0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + 1.375y + y2 = 0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + 1.375y + y2 = 0 + 0.1666666667 1.375y + y2 = 0 + 0.1666666667 Combine like terms: 0 + 0.1666666667 = 0.1666666667 1.375y + y2 = 0.1666666667 The y term is 1.375y. Take half its coefficient (0.6875). Square it (0.47265625) and add it to both sides. Add '0.47265625' to each side of the equation. 1.375y + 0.47265625 + y2 = 0.1666666667 + 0.47265625 Reorder the terms: 0.47265625 + 1.375y + y2 = 0.1666666667 + 0.47265625 Combine like terms: 0.1666666667 + 0.47265625 = 0.6393229167 0.47265625 + 1.375y + y2 = 0.6393229167 Factor a perfect square on the left side: (y + 0.6875)(y + 0.6875) = 0.6393229167 Calculate the square root of the right side: 0.799576711 Break this problem into two subproblems by setting (y + 0.6875) equal to 0.799576711 and -0.799576711.Subproblem 1
y + 0.6875 = 0.799576711 Simplifying y + 0.6875 = 0.799576711 Reorder the terms: 0.6875 + y = 0.799576711 Solving 0.6875 + y = 0.799576711 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + y = 0.799576711 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + y = 0.799576711 + -0.6875 y = 0.799576711 + -0.6875 Combine like terms: 0.799576711 + -0.6875 = 0.112076711 y = 0.112076711 Simplifying y = 0.112076711Subproblem 2
y + 0.6875 = -0.799576711 Simplifying y + 0.6875 = -0.799576711 Reorder the terms: 0.6875 + y = -0.799576711 Solving 0.6875 + y = -0.799576711 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + y = -0.799576711 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + y = -0.799576711 + -0.6875 y = -0.799576711 + -0.6875 Combine like terms: -0.799576711 + -0.6875 = -1.487076711 y = -1.487076711 Simplifying y = -1.487076711Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.112076711, -1.487076711}
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