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Simplifying 5(y + -4)(y + 5) + 3(y + 3)(y + 5) = 7(y + 3)(y + -4) Reorder the terms: 5(-4 + y)(y + 5) + 3(y + 3)(y + 5) = 7(y + 3)(y + -4) Reorder the terms: 5(-4 + y)(5 + y) + 3(y + 3)(y + 5) = 7(y + 3)(y + -4) Multiply (-4 + y) * (5 + y) 5(-4(5 + y) + y(5 + y)) + 3(y + 3)(y + 5) = 7(y + 3)(y + -4) 5((5 * -4 + y * -4) + y(5 + y)) + 3(y + 3)(y + 5) = 7(y + 3)(y + -4) 5((-20 + -4y) + y(5 + y)) + 3(y + 3)(y + 5) = 7(y + 3)(y + -4) 5(-20 + -4y + (5 * y + y * y)) + 3(y + 3)(y + 5) = 7(y + 3)(y + -4) 5(-20 + -4y + (5y + y2)) + 3(y + 3)(y + 5) = 7(y + 3)(y + -4) Combine like terms: -4y + 5y = 1y 5(-20 + 1y + y2) + 3(y + 3)(y + 5) = 7(y + 3)(y + -4) (-20 * 5 + 1y * 5 + y2 * 5) + 3(y + 3)(y + 5) = 7(y + 3)(y + -4) (-100 + 5y + 5y2) + 3(y + 3)(y + 5) = 7(y + 3)(y + -4) Reorder the terms: -100 + 5y + 5y2 + 3(3 + y)(y + 5) = 7(y + 3)(y + -4) Reorder the terms: -100 + 5y + 5y2 + 3(3 + y)(5 + y) = 7(y + 3)(y + -4) Multiply (3 + y) * (5 + y) -100 + 5y + 5y2 + 3(3(5 + y) + y(5 + y)) = 7(y + 3)(y + -4) -100 + 5y + 5y2 + 3((5 * 3 + y * 3) + y(5 + y)) = 7(y + 3)(y + -4) -100 + 5y + 5y2 + 3((15 + 3y) + y(5 + y)) = 7(y + 3)(y + -4) -100 + 5y + 5y2 + 3(15 + 3y + (5 * y + y * y)) = 7(y + 3)(y + -4) -100 + 5y + 5y2 + 3(15 + 3y + (5y + y2)) = 7(y + 3)(y + -4) Combine like terms: 3y + 5y = 8y -100 + 5y + 5y2 + 3(15 + 8y + y2) = 7(y + 3)(y + -4) -100 + 5y + 5y2 + (15 * 3 + 8y * 3 + y2 * 3) = 7(y + 3)(y + -4) -100 + 5y + 5y2 + (45 + 24y + 3y2) = 7(y + 3)(y + -4) Reorder the terms: -100 + 45 + 5y + 24y + 5y2 + 3y2 = 7(y + 3)(y + -4) Combine like terms: -100 + 45 = -55 -55 + 5y + 24y + 5y2 + 3y2 = 7(y + 3)(y + -4) Combine like terms: 5y + 24y = 29y -55 + 29y + 5y2 + 3y2 = 7(y + 3)(y + -4) Combine like terms: 5y2 + 3y2 = 8y2 -55 + 29y + 8y2 = 7(y + 3)(y + -4) Reorder the terms: -55 + 29y + 8y2 = 7(3 + y)(y + -4) Reorder the terms: -55 + 29y + 8y2 = 7(3 + y)(-4 + y) Multiply (3 + y) * (-4 + y) -55 + 29y + 8y2 = 7(3(-4 + y) + y(-4 + y)) -55 + 29y + 8y2 = 7((-4 * 3 + y * 3) + y(-4 + y)) -55 + 29y + 8y2 = 7((-12 + 3y) + y(-4 + y)) -55 + 29y + 8y2 = 7(-12 + 3y + (-4 * y + y * y)) -55 + 29y + 8y2 = 7(-12 + 3y + (-4y + y2)) Combine like terms: 3y + -4y = -1y -55 + 29y + 8y2 = 7(-12 + -1y + y2) -55 + 29y + 8y2 = (-12 * 7 + -1y * 7 + y2 * 7) -55 + 29y + 8y2 = (-84 + -7y + 7y2) Solving -55 + 29y + 8y2 = -84 + -7y + 7y2 Solving for variable 'y'. Reorder the terms: -55 + 84 + 29y + 7y + 8y2 + -7y2 = -84 + -7y + 7y2 + 84 + 7y + -7y2 Combine like terms: -55 + 84 = 29 29 + 29y + 7y + 8y2 + -7y2 = -84 + -7y + 7y2 + 84 + 7y + -7y2 Combine like terms: 29y + 7y = 36y 29 + 36y + 8y2 + -7y2 = -84 + -7y + 7y2 + 84 + 7y + -7y2 Combine like terms: 8y2 + -7y2 = 1y2 29 + 36y + 1y2 = -84 + -7y + 7y2 + 84 + 7y + -7y2 Reorder the terms: 29 + 36y + 1y2 = -84 + 84 + -7y + 7y + 7y2 + -7y2 Combine like terms: -84 + 84 = 0 29 + 36y + 1y2 = 0 + -7y + 7y + 7y2 + -7y2 29 + 36y + 1y2 = -7y + 7y + 7y2 + -7y2 Combine like terms: -7y + 7y = 0 29 + 36y + 1y2 = 0 + 7y2 + -7y2 29 + 36y + 1y2 = 7y2 + -7y2 Combine like terms: 7y2 + -7y2 = 0 29 + 36y + 1y2 = 0 Begin completing the square. Move the constant term to the right: Add '-29' to each side of the equation. 29 + 36y + -29 + y2 = 0 + -29 Reorder the terms: 29 + -29 + 36y + y2 = 0 + -29 Combine like terms: 29 + -29 = 0 0 + 36y + y2 = 0 + -29 36y + y2 = 0 + -29 Combine like terms: 0 + -29 = -29 36y + y2 = -29 The y term is 36y. Take half its coefficient (18). Square it (324) and add it to both sides. Add '324' to each side of the equation. 36y + 324 + y2 = -29 + 324 Reorder the terms: 324 + 36y + y2 = -29 + 324 Combine like terms: -29 + 324 = 295 324 + 36y + y2 = 295 Factor a perfect square on the left side: (y + 18)(y + 18) = 295 Calculate the square root of the right side: 17.175564037 Break this problem into two subproblems by setting (y + 18) equal to 17.175564037 and -17.175564037.Subproblem 1
y + 18 = 17.175564037 Simplifying y + 18 = 17.175564037 Reorder the terms: 18 + y = 17.175564037 Solving 18 + y = 17.175564037 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-18' to each side of the equation. 18 + -18 + y = 17.175564037 + -18 Combine like terms: 18 + -18 = 0 0 + y = 17.175564037 + -18 y = 17.175564037 + -18 Combine like terms: 17.175564037 + -18 = -0.824435963 y = -0.824435963 Simplifying y = -0.824435963Subproblem 2
y + 18 = -17.175564037 Simplifying y + 18 = -17.175564037 Reorder the terms: 18 + y = -17.175564037 Solving 18 + y = -17.175564037 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-18' to each side of the equation. 18 + -18 + y = -17.175564037 + -18 Combine like terms: 18 + -18 = 0 0 + y = -17.175564037 + -18 y = -17.175564037 + -18 Combine like terms: -17.175564037 + -18 = -35.175564037 y = -35.175564037 Simplifying y = -35.175564037Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.824435963, -35.175564037}
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