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Simplifying 26 + -1[5 + 4y + -2(y + 2)] = -3(4y + -5)[-6(y + -1) + -12y + 4] Reorder the terms: 26 + -1[5 + 4y + -2(2 + y)] = -3(4y + -5)[-6(y + -1) + -12y + 4] 26 + -1[5 + 4y + (2 * -2 + y * -2)] = -3(4y + -5)[-6(y + -1) + -12y + 4] 26 + -1[5 + 4y + (-4 + -2y)] = -3(4y + -5)[-6(y + -1) + -12y + 4] Reorder the terms: 26 + -1[5 + -4 + 4y + -2y] = -3(4y + -5)[-6(y + -1) + -12y + 4] Combine like terms: 5 + -4 = 1 26 + -1[1 + 4y + -2y] = -3(4y + -5)[-6(y + -1) + -12y + 4] Combine like terms: 4y + -2y = 2y 26 + -1[1 + 2y] = -3(4y + -5)[-6(y + -1) + -12y + 4] 26 + [1 * -1 + 2y * -1] = -3(4y + -5)[-6(y + -1) + -12y + 4] 26 + [-1 + -2y] = -3(4y + -5)[-6(y + -1) + -12y + 4] Combine like terms: 26 + -1 = 25 25 + -2y = -3(4y + -5)[-6(y + -1) + -12y + 4] Reorder the terms: 25 + -2y = -3(-5 + 4y)[-6(y + -1) + -12y + 4] Reorder the terms: 25 + -2y = -3(-5 + 4y)[-6(-1 + y) + -12y + 4] 25 + -2y = -3(-5 + 4y)[(-1 * -6 + y * -6) + -12y + 4] 25 + -2y = -3(-5 + 4y)[(6 + -6y) + -12y + 4] Reorder the terms: 25 + -2y = -3(-5 + 4y)[6 + 4 + -6y + -12y] Combine like terms: 6 + 4 = 10 25 + -2y = -3(-5 + 4y)[10 + -6y + -12y] Combine like terms: -6y + -12y = -18y 25 + -2y = -3(-5 + 4y)[10 + -18y] Multiply (-5 + 4y) * [10 + -18y] 25 + -2y = -3(-5[10 + -18y] + 4y * [10 + -18y]) 25 + -2y = -3([10 * -5 + -18y * -5] + 4y * [10 + -18y]) 25 + -2y = -3([-50 + 90y] + 4y * [10 + -18y]) 25 + -2y = -3(-50 + 90y + [10 * 4y + -18y * 4y]) 25 + -2y = -3(-50 + 90y + [40y + -72y2]) Combine like terms: 90y + 40y = 130y 25 + -2y = -3(-50 + 130y + -72y2) 25 + -2y = (-50 * -3 + 130y * -3 + -72y2 * -3) 25 + -2y = (150 + -390y + 216y2) Solving 25 + -2y = 150 + -390y + 216y2 Solving for variable 'y'. Reorder the terms: 25 + -150 + -2y + 390y + -216y2 = 150 + -390y + 216y2 + -150 + 390y + -216y2 Combine like terms: 25 + -150 = -125 -125 + -2y + 390y + -216y2 = 150 + -390y + 216y2 + -150 + 390y + -216y2 Combine like terms: -2y + 390y = 388y -125 + 388y + -216y2 = 150 + -390y + 216y2 + -150 + 390y + -216y2 Reorder the terms: -125 + 388y + -216y2 = 150 + -150 + -390y + 390y + 216y2 + -216y2 Combine like terms: 150 + -150 = 0 -125 + 388y + -216y2 = 0 + -390y + 390y + 216y2 + -216y2 -125 + 388y + -216y2 = -390y + 390y + 216y2 + -216y2 Combine like terms: -390y + 390y = 0 -125 + 388y + -216y2 = 0 + 216y2 + -216y2 -125 + 388y + -216y2 = 216y2 + -216y2 Combine like terms: 216y2 + -216y2 = 0 -125 + 388y + -216y2 = 0 Begin completing the square. Divide all terms by -216 the coefficient of the squared term: Divide each side by '-216'. 0.5787037037 + -1.796296296y + y2 = 0 Move the constant term to the right: Add '-0.5787037037' to each side of the equation. 0.5787037037 + -1.796296296y + -0.5787037037 + y2 = 0 + -0.5787037037 Reorder the terms: 0.5787037037 + -0.5787037037 + -1.796296296y + y2 = 0 + -0.5787037037 Combine like terms: 0.5787037037 + -0.5787037037 = 0.0000000000 0.0000000000 + -1.796296296y + y2 = 0 + -0.5787037037 -1.796296296y + y2 = 0 + -0.5787037037 Combine like terms: 0 + -0.5787037037 = -0.5787037037 -1.796296296y + y2 = -0.5787037037 The y term is -1.796296296y. Take half its coefficient (-0.898148148). Square it (0.8066700958) and add it to both sides. Add '0.8066700958' to each side of the equation. -1.796296296y + 0.8066700958 + y2 = -0.5787037037 + 0.8066700958 Reorder the terms: 0.8066700958 + -1.796296296y + y2 = -0.5787037037 + 0.8066700958 Combine like terms: -0.5787037037 + 0.8066700958 = 0.2279663921 0.8066700958 + -1.796296296y + y2 = 0.2279663921 Factor a perfect square on the left side: (y + -0.898148148)(y + -0.898148148) = 0.2279663921 Calculate the square root of the right side: 0.477458262 Break this problem into two subproblems by setting (y + -0.898148148) equal to 0.477458262 and -0.477458262.Subproblem 1
y + -0.898148148 = 0.477458262 Simplifying y + -0.898148148 = 0.477458262 Reorder the terms: -0.898148148 + y = 0.477458262 Solving -0.898148148 + y = 0.477458262 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.898148148' to each side of the equation. -0.898148148 + 0.898148148 + y = 0.477458262 + 0.898148148 Combine like terms: -0.898148148 + 0.898148148 = 0.000000000 0.000000000 + y = 0.477458262 + 0.898148148 y = 0.477458262 + 0.898148148 Combine like terms: 0.477458262 + 0.898148148 = 1.37560641 y = 1.37560641 Simplifying y = 1.37560641Subproblem 2
y + -0.898148148 = -0.477458262 Simplifying y + -0.898148148 = -0.477458262 Reorder the terms: -0.898148148 + y = -0.477458262 Solving -0.898148148 + y = -0.477458262 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.898148148' to each side of the equation. -0.898148148 + 0.898148148 + y = -0.477458262 + 0.898148148 Combine like terms: -0.898148148 + 0.898148148 = 0.000000000 0.000000000 + y = -0.477458262 + 0.898148148 y = -0.477458262 + 0.898148148 Combine like terms: -0.477458262 + 0.898148148 = 0.420689886 y = 0.420689886 Simplifying y = 0.420689886Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.37560641, 0.420689886}
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