f(0)=-5x^2+3x+9

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Solution for f(0)=-5x^2+3x+9 equation:


Simplifying
f(0) = -5x2 + 3x + 9

Reorder the terms for easier multiplication:
0f = -5x2 + 3x + 9

Anything times zero is zero.
0f = -5x2 + 3x + 9

Reorder the terms:
0 = 9 + 3x + -5x2

Solving
0 = 9 + 3x + -5x2

Solving for variable 'x'.

Combine like terms: 0 + -9 = -9
-9 + -3x + 5x2 = 9 + 3x + -5x2 + -9 + -3x + 5x2

Reorder the terms:
-9 + -3x + 5x2 = 9 + -9 + 3x + -3x + -5x2 + 5x2

Combine like terms: 9 + -9 = 0
-9 + -3x + 5x2 = 0 + 3x + -3x + -5x2 + 5x2
-9 + -3x + 5x2 = 3x + -3x + -5x2 + 5x2

Combine like terms: 3x + -3x = 0
-9 + -3x + 5x2 = 0 + -5x2 + 5x2
-9 + -3x + 5x2 = -5x2 + 5x2

Combine like terms: -5x2 + 5x2 = 0
-9 + -3x + 5x2 = 0

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

Divide each side by '5'.
-1.8 + -0.6x + x2 = 0

Move the constant term to the right:

Add '1.8' to each side of the equation.
-1.8 + -0.6x + 1.8 + x2 = 0 + 1.8

Reorder the terms:
-1.8 + 1.8 + -0.6x + x2 = 0 + 1.8

Combine like terms: -1.8 + 1.8 = 0.0
0.0 + -0.6x + x2 = 0 + 1.8
-0.6x + x2 = 0 + 1.8

Combine like terms: 0 + 1.8 = 1.8
-0.6x + x2 = 1.8

The x term is -0.6x.  Take half its coefficient (-0.3).
Square it (0.09) and add it to both sides.

Add '0.09' to each side of the equation.
-0.6x + 0.09 + x2 = 1.8 + 0.09

Reorder the terms:
0.09 + -0.6x + x2 = 1.8 + 0.09

Combine like terms: 1.8 + 0.09 = 1.89
0.09 + -0.6x + x2 = 1.89

Factor a perfect square on the left side:
(x + -0.3)(x + -0.3) = 1.89

Calculate the square root of the right side: 1.374772708

Break this problem into two subproblems by setting 
(x + -0.3) equal to 1.374772708 and -1.374772708.

Subproblem 1

x + -0.3 = 1.374772708 Simplifying x + -0.3 = 1.374772708 Reorder the terms: -0.3 + x = 1.374772708 Solving -0.3 + x = 1.374772708 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + x = 1.374772708 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + x = 1.374772708 + 0.3 x = 1.374772708 + 0.3 Combine like terms: 1.374772708 + 0.3 = 1.674772708 x = 1.674772708 Simplifying x = 1.674772708

Subproblem 2

x + -0.3 = -1.374772708 Simplifying x + -0.3 = -1.374772708 Reorder the terms: -0.3 + x = -1.374772708 Solving -0.3 + x = -1.374772708 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + x = -1.374772708 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + x = -1.374772708 + 0.3 x = -1.374772708 + 0.3 Combine like terms: -1.374772708 + 0.3 = -1.074772708 x = -1.074772708 Simplifying x = -1.074772708

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.674772708, -1.074772708}

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