5(y-4)(y+5)+3(y+3)(y+5)=7(y+3)(y-4)

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Solution for 5(y-4)(y+5)+3(y+3)(y+5)=7(y+3)(y-4) equation:


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 = -0.824435963

Subproblem 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 = -35.175564037

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-0.824435963, -35.175564037}

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