x=-15x^2+60x+40

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Solution for x=-15x^2+60x+40 equation:


Simplifying
x = -15x2 + 60x + 40

Reorder the terms:
x = 40 + 60x + -15x2

Solving
x = 40 + 60x + -15x2

Solving for variable 'x'.

Reorder the terms:
-40 + x + -60x + 15x2 = 40 + 60x + -15x2 + -40 + -60x + 15x2

Combine like terms: x + -60x = -59x
-40 + -59x + 15x2 = 40 + 60x + -15x2 + -40 + -60x + 15x2

Reorder the terms:
-40 + -59x + 15x2 = 40 + -40 + 60x + -60x + -15x2 + 15x2

Combine like terms: 40 + -40 = 0
-40 + -59x + 15x2 = 0 + 60x + -60x + -15x2 + 15x2
-40 + -59x + 15x2 = 60x + -60x + -15x2 + 15x2

Combine like terms: 60x + -60x = 0
-40 + -59x + 15x2 = 0 + -15x2 + 15x2
-40 + -59x + 15x2 = -15x2 + 15x2

Combine like terms: -15x2 + 15x2 = 0
-40 + -59x + 15x2 = 0

Begin completing the square.  Divide all terms by
15 the coefficient of the squared term: 

Divide each side by '15'.
-2.666666667 + -3.933333333x + x2 = 0

Move the constant term to the right:

Add '2.666666667' to each side of the equation.
-2.666666667 + -3.933333333x + 2.666666667 + x2 = 0 + 2.666666667

Reorder the terms:
-2.666666667 + 2.666666667 + -3.933333333x + x2 = 0 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + -3.933333333x + x2 = 0 + 2.666666667
-3.933333333x + x2 = 0 + 2.666666667

Combine like terms: 0 + 2.666666667 = 2.666666667
-3.933333333x + x2 = 2.666666667

The x term is -3.933333333x.  Take half its coefficient (-1.966666667).
Square it (3.867777779) and add it to both sides.

Add '3.867777779' to each side of the equation.
-3.933333333x + 3.867777779 + x2 = 2.666666667 + 3.867777779

Reorder the terms:
3.867777779 + -3.933333333x + x2 = 2.666666667 + 3.867777779

Combine like terms: 2.666666667 + 3.867777779 = 6.534444446
3.867777779 + -3.933333333x + x2 = 6.534444446

Factor a perfect square on the left side:
(x + -1.966666667)(x + -1.966666667) = 6.534444446

Calculate the square root of the right side: 2.556255943

Break this problem into two subproblems by setting 
(x + -1.966666667) equal to 2.556255943 and -2.556255943.

Subproblem 1

x + -1.966666667 = 2.556255943 Simplifying x + -1.966666667 = 2.556255943 Reorder the terms: -1.966666667 + x = 2.556255943 Solving -1.966666667 + x = 2.556255943 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.966666667' to each side of the equation. -1.966666667 + 1.966666667 + x = 2.556255943 + 1.966666667 Combine like terms: -1.966666667 + 1.966666667 = 0.000000000 0.000000000 + x = 2.556255943 + 1.966666667 x = 2.556255943 + 1.966666667 Combine like terms: 2.556255943 + 1.966666667 = 4.52292261 x = 4.52292261 Simplifying x = 4.52292261

Subproblem 2

x + -1.966666667 = -2.556255943 Simplifying x + -1.966666667 = -2.556255943 Reorder the terms: -1.966666667 + x = -2.556255943 Solving -1.966666667 + x = -2.556255943 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.966666667' to each side of the equation. -1.966666667 + 1.966666667 + x = -2.556255943 + 1.966666667 Combine like terms: -1.966666667 + 1.966666667 = 0.000000000 0.000000000 + x = -2.556255943 + 1.966666667 x = -2.556255943 + 1.966666667 Combine like terms: -2.556255943 + 1.966666667 = -0.589589276 x = -0.589589276 Simplifying x = -0.589589276

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.52292261, -0.589589276}

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