4x^2-22x+7=0

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Solution for 4x^2-22x+7=0 equation:


Simplifying
4x2 + -22x + 7 = 0

Reorder the terms:
7 + -22x + 4x2 = 0

Solving
7 + -22x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
1.75 + -5.5x + x2 = 0

Move the constant term to the right:

Add '-1.75' to each side of the equation.
1.75 + -5.5x + -1.75 + x2 = 0 + -1.75

Reorder the terms:
1.75 + -1.75 + -5.5x + x2 = 0 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + -5.5x + x2 = 0 + -1.75
-5.5x + x2 = 0 + -1.75

Combine like terms: 0 + -1.75 = -1.75
-5.5x + x2 = -1.75

The x term is -5.5x.  Take half its coefficient (-2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
-5.5x + 7.5625 + x2 = -1.75 + 7.5625

Reorder the terms:
7.5625 + -5.5x + x2 = -1.75 + 7.5625

Combine like terms: -1.75 + 7.5625 = 5.8125
7.5625 + -5.5x + x2 = 5.8125

Factor a perfect square on the left side:
(x + -2.75)(x + -2.75) = 5.8125

Calculate the square root of the right side: 2.41091269

Break this problem into two subproblems by setting 
(x + -2.75) equal to 2.41091269 and -2.41091269.

Subproblem 1

x + -2.75 = 2.41091269 Simplifying x + -2.75 = 2.41091269 Reorder the terms: -2.75 + x = 2.41091269 Solving -2.75 + x = 2.41091269 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + x = 2.41091269 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + x = 2.41091269 + 2.75 x = 2.41091269 + 2.75 Combine like terms: 2.41091269 + 2.75 = 5.16091269 x = 5.16091269 Simplifying x = 5.16091269

Subproblem 2

x + -2.75 = -2.41091269 Simplifying x + -2.75 = -2.41091269 Reorder the terms: -2.75 + x = -2.41091269 Solving -2.75 + x = -2.41091269 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + x = -2.41091269 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + x = -2.41091269 + 2.75 x = -2.41091269 + 2.75 Combine like terms: -2.41091269 + 2.75 = 0.33908731 x = 0.33908731 Simplifying x = 0.33908731

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.16091269, 0.33908731}

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