f(0)=-8x^2+9x+9

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Solution for f(0)=-8x^2+9x+9 equation:


Simplifying
f(0) = -8x2 + 9x + 9

Reorder the terms for easier multiplication:
0f = -8x2 + 9x + 9

Anything times zero is zero.
0f = -8x2 + 9x + 9

Reorder the terms:
0 = 9 + 9x + -8x2

Solving
0 = 9 + 9x + -8x2

Solving for variable 'x'.

Combine like terms: 0 + -9 = -9
-9 + -9x + 8x2 = 9 + 9x + -8x2 + -9 + -9x + 8x2

Reorder the terms:
-9 + -9x + 8x2 = 9 + -9 + 9x + -9x + -8x2 + 8x2

Combine like terms: 9 + -9 = 0
-9 + -9x + 8x2 = 0 + 9x + -9x + -8x2 + 8x2
-9 + -9x + 8x2 = 9x + -9x + -8x2 + 8x2

Combine like terms: 9x + -9x = 0
-9 + -9x + 8x2 = 0 + -8x2 + 8x2
-9 + -9x + 8x2 = -8x2 + 8x2

Combine like terms: -8x2 + 8x2 = 0
-9 + -9x + 8x2 = 0

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

Divide each side by '8'.
-1.125 + -1.125x + x2 = 0

Move the constant term to the right:

Add '1.125' to each side of the equation.
-1.125 + -1.125x + 1.125 + x2 = 0 + 1.125

Reorder the terms:
-1.125 + 1.125 + -1.125x + x2 = 0 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + -1.125x + x2 = 0 + 1.125
-1.125x + x2 = 0 + 1.125

Combine like terms: 0 + 1.125 = 1.125
-1.125x + x2 = 1.125

The x term is -1.125x.  Take half its coefficient (-0.5625).
Square it (0.31640625) and add it to both sides.

Add '0.31640625' to each side of the equation.
-1.125x + 0.31640625 + x2 = 1.125 + 0.31640625

Reorder the terms:
0.31640625 + -1.125x + x2 = 1.125 + 0.31640625

Combine like terms: 1.125 + 0.31640625 = 1.44140625
0.31640625 + -1.125x + x2 = 1.44140625

Factor a perfect square on the left side:
(x + -0.5625)(x + -0.5625) = 1.44140625

Calculate the square root of the right side: 1.200585795

Break this problem into two subproblems by setting 
(x + -0.5625) equal to 1.200585795 and -1.200585795.

Subproblem 1

x + -0.5625 = 1.200585795 Simplifying x + -0.5625 = 1.200585795 Reorder the terms: -0.5625 + x = 1.200585795 Solving -0.5625 + x = 1.200585795 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + x = 1.200585795 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + x = 1.200585795 + 0.5625 x = 1.200585795 + 0.5625 Combine like terms: 1.200585795 + 0.5625 = 1.763085795 x = 1.763085795 Simplifying x = 1.763085795

Subproblem 2

x + -0.5625 = -1.200585795 Simplifying x + -0.5625 = -1.200585795 Reorder the terms: -0.5625 + x = -1.200585795 Solving -0.5625 + x = -1.200585795 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + x = -1.200585795 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + x = -1.200585795 + 0.5625 x = -1.200585795 + 0.5625 Combine like terms: -1.200585795 + 0.5625 = -0.638085795 x = -0.638085795 Simplifying x = -0.638085795

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.763085795, -0.638085795}

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