f(0)=3x^2+2x-4

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Solution for f(0)=3x^2+2x-4 equation:


Simplifying
f(0) = 3x2 + 2x + -4

Reorder the terms for easier multiplication:
0f = 3x2 + 2x + -4

Anything times zero is zero.
0f = 3x2 + 2x + -4

Reorder the terms:
0 = -4 + 2x + 3x2

Solving
0 = -4 + 2x + 3x2

Solving for variable 'x'.

Combine like terms: 0 + 4 = 4
4 + -2x + -3x2 = -4 + 2x + 3x2 + 4 + -2x + -3x2

Reorder the terms:
4 + -2x + -3x2 = -4 + 4 + 2x + -2x + 3x2 + -3x2

Combine like terms: -4 + 4 = 0
4 + -2x + -3x2 = 0 + 2x + -2x + 3x2 + -3x2
4 + -2x + -3x2 = 2x + -2x + 3x2 + -3x2

Combine like terms: 2x + -2x = 0
4 + -2x + -3x2 = 0 + 3x2 + -3x2
4 + -2x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
4 + -2x + -3x2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-1.333333333 + 0.6666666667x + x2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + 0.6666666667x + 1.333333333 + x2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + 0.6666666667x + x2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + 0.6666666667x + x2 = 0 + 1.333333333
0.6666666667x + x2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
0.6666666667x + x2 = 1.333333333

The x term is 0.6666666667x.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667x + 0.1111111112 + x2 = 1.333333333 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 1.333333333 + 0.1111111112

Combine like terms: 1.333333333 + 0.1111111112 = 1.4444444442
0.1111111112 + 0.6666666667x + x2 = 1.4444444442

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 1.4444444442

Calculate the square root of the right side: 1.201850425

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 1.201850425 and -1.201850425.

Subproblem 1

x + 0.3333333334 = 1.201850425 Simplifying x + 0.3333333334 = 1.201850425 Reorder the terms: 0.3333333334 + x = 1.201850425 Solving 0.3333333334 + x = 1.201850425 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = 1.201850425 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 1.201850425 + -0.3333333334 x = 1.201850425 + -0.3333333334 Combine like terms: 1.201850425 + -0.3333333334 = 0.8685170916 x = 0.8685170916 Simplifying x = 0.8685170916

Subproblem 2

x + 0.3333333334 = -1.201850425 Simplifying x + 0.3333333334 = -1.201850425 Reorder the terms: 0.3333333334 + x = -1.201850425 Solving 0.3333333334 + x = -1.201850425 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = -1.201850425 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -1.201850425 + -0.3333333334 x = -1.201850425 + -0.3333333334 Combine like terms: -1.201850425 + -0.3333333334 = -1.5351837584 x = -1.5351837584 Simplifying x = -1.5351837584

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.8685170916, -1.5351837584}

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