21x^2-78x+49=0

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Solution for 21x^2-78x+49=0 equation:


Simplifying
21x2 + -78x + 49 = 0

Reorder the terms:
49 + -78x + 21x2 = 0

Solving
49 + -78x + 21x2 = 0

Solving for variable 'x'.

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

Divide each side by '21'.
2.333333333 + -3.714285714x + x2 = 0

Move the constant term to the right:

Add '-2.333333333' to each side of the equation.
2.333333333 + -3.714285714x + -2.333333333 + x2 = 0 + -2.333333333

Reorder the terms:
2.333333333 + -2.333333333 + -3.714285714x + x2 = 0 + -2.333333333

Combine like terms: 2.333333333 + -2.333333333 = 0.000000000
0.000000000 + -3.714285714x + x2 = 0 + -2.333333333
-3.714285714x + x2 = 0 + -2.333333333

Combine like terms: 0 + -2.333333333 = -2.333333333
-3.714285714x + x2 = -2.333333333

The x term is -3.714285714x.  Take half its coefficient (-1.857142857).
Square it (3.448979591) and add it to both sides.

Add '3.448979591' to each side of the equation.
-3.714285714x + 3.448979591 + x2 = -2.333333333 + 3.448979591

Reorder the terms:
3.448979591 + -3.714285714x + x2 = -2.333333333 + 3.448979591

Combine like terms: -2.333333333 + 3.448979591 = 1.115646258
3.448979591 + -3.714285714x + x2 = 1.115646258

Factor a perfect square on the left side:
(x + -1.857142857)(x + -1.857142857) = 1.115646258

Calculate the square root of the right side: 1.056241572

Break this problem into two subproblems by setting 
(x + -1.857142857) equal to 1.056241572 and -1.056241572.

Subproblem 1

x + -1.857142857 = 1.056241572 Simplifying x + -1.857142857 = 1.056241572 Reorder the terms: -1.857142857 + x = 1.056241572 Solving -1.857142857 + x = 1.056241572 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.857142857' to each side of the equation. -1.857142857 + 1.857142857 + x = 1.056241572 + 1.857142857 Combine like terms: -1.857142857 + 1.857142857 = 0.000000000 0.000000000 + x = 1.056241572 + 1.857142857 x = 1.056241572 + 1.857142857 Combine like terms: 1.056241572 + 1.857142857 = 2.913384429 x = 2.913384429 Simplifying x = 2.913384429

Subproblem 2

x + -1.857142857 = -1.056241572 Simplifying x + -1.857142857 = -1.056241572 Reorder the terms: -1.857142857 + x = -1.056241572 Solving -1.857142857 + x = -1.056241572 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.857142857' to each side of the equation. -1.857142857 + 1.857142857 + x = -1.056241572 + 1.857142857 Combine like terms: -1.857142857 + 1.857142857 = 0.000000000 0.000000000 + x = -1.056241572 + 1.857142857 x = -1.056241572 + 1.857142857 Combine like terms: -1.056241572 + 1.857142857 = 0.800901285 x = 0.800901285 Simplifying x = 0.800901285

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.913384429, 0.800901285}

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