4f^2+6f-9=0

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Solution for 4f^2+6f-9=0 equation:


Simplifying
4f2 + 6f + -9 = 0

Reorder the terms:
-9 + 6f + 4f2 = 0

Solving
-9 + 6f + 4f2 = 0

Solving for variable 'f'.

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

Divide each side by '4'.
-2.25 + 1.5f + f2 = 0

Move the constant term to the right:

Add '2.25' to each side of the equation.
-2.25 + 1.5f + 2.25 + f2 = 0 + 2.25

Reorder the terms:
-2.25 + 2.25 + 1.5f + f2 = 0 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + 1.5f + f2 = 0 + 2.25
1.5f + f2 = 0 + 2.25

Combine like terms: 0 + 2.25 = 2.25
1.5f + f2 = 2.25

The f term is 1.5f.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5f + 0.5625 + f2 = 2.25 + 0.5625

Reorder the terms:
0.5625 + 1.5f + f2 = 2.25 + 0.5625

Combine like terms: 2.25 + 0.5625 = 2.8125
0.5625 + 1.5f + f2 = 2.8125

Factor a perfect square on the left side:
(f + 0.75)(f + 0.75) = 2.8125

Calculate the square root of the right side: 1.677050983

Break this problem into two subproblems by setting 
(f + 0.75) equal to 1.677050983 and -1.677050983.

Subproblem 1

f + 0.75 = 1.677050983 Simplifying f + 0.75 = 1.677050983 Reorder the terms: 0.75 + f = 1.677050983 Solving 0.75 + f = 1.677050983 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + f = 1.677050983 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + f = 1.677050983 + -0.75 f = 1.677050983 + -0.75 Combine like terms: 1.677050983 + -0.75 = 0.927050983 f = 0.927050983 Simplifying f = 0.927050983

Subproblem 2

f + 0.75 = -1.677050983 Simplifying f + 0.75 = -1.677050983 Reorder the terms: 0.75 + f = -1.677050983 Solving 0.75 + f = -1.677050983 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + f = -1.677050983 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + f = -1.677050983 + -0.75 f = -1.677050983 + -0.75 Combine like terms: -1.677050983 + -0.75 = -2.427050983 f = -2.427050983 Simplifying f = -2.427050983

Solution

The solution to the problem is based on the solutions from the subproblems. f = {0.927050983, -2.427050983}

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