4x^2=14x-9

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Solution for 4x^2=14x-9 equation:


Simplifying
4x2 = 14x + -9

Reorder the terms:
4x2 = -9 + 14x

Solving
4x2 = -9 + 14x

Solving for variable 'x'.

Reorder the terms:
9 + -14x + 4x2 = -9 + 14x + 9 + -14x

Reorder the terms:
9 + -14x + 4x2 = -9 + 9 + 14x + -14x

Combine like terms: -9 + 9 = 0
9 + -14x + 4x2 = 0 + 14x + -14x
9 + -14x + 4x2 = 14x + -14x

Combine like terms: 14x + -14x = 0
9 + -14x + 4x2 = 0

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

Divide each side by '4'.
2.25 + -3.5x + x2 = 0

Move the constant term to the right:

Add '-2.25' to each side of the equation.
2.25 + -3.5x + -2.25 + x2 = 0 + -2.25

Reorder the terms:
2.25 + -2.25 + -3.5x + x2 = 0 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + -3.5x + x2 = 0 + -2.25
-3.5x + x2 = 0 + -2.25

Combine like terms: 0 + -2.25 = -2.25
-3.5x + x2 = -2.25

The x term is -3.5x.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5x + 3.0625 + x2 = -2.25 + 3.0625

Reorder the terms:
3.0625 + -3.5x + x2 = -2.25 + 3.0625

Combine like terms: -2.25 + 3.0625 = 0.8125
3.0625 + -3.5x + x2 = 0.8125

Factor a perfect square on the left side:
(x + -1.75)(x + -1.75) = 0.8125

Calculate the square root of the right side: 0.901387819

Break this problem into two subproblems by setting 
(x + -1.75) equal to 0.901387819 and -0.901387819.

Subproblem 1

x + -1.75 = 0.901387819 Simplifying x + -1.75 = 0.901387819 Reorder the terms: -1.75 + x = 0.901387819 Solving -1.75 + x = 0.901387819 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + x = 0.901387819 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + x = 0.901387819 + 1.75 x = 0.901387819 + 1.75 Combine like terms: 0.901387819 + 1.75 = 2.651387819 x = 2.651387819 Simplifying x = 2.651387819

Subproblem 2

x + -1.75 = -0.901387819 Simplifying x + -1.75 = -0.901387819 Reorder the terms: -1.75 + x = -0.901387819 Solving -1.75 + x = -0.901387819 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + x = -0.901387819 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + x = -0.901387819 + 1.75 x = -0.901387819 + 1.75 Combine like terms: -0.901387819 + 1.75 = 0.848612181 x = 0.848612181 Simplifying x = 0.848612181

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.651387819, 0.848612181}

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