21x^2-97x+63=0

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Solution for 21x^2-97x+63=0 equation:


Simplifying
21x2 + -97x + 63 = 0

Reorder the terms:
63 + -97x + 21x2 = 0

Solving
63 + -97x + 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'.
3 + -4.619047619x + x2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -4.619047619x + -3 + x2 = 0 + -3

Reorder the terms:
3 + -3 + -4.619047619x + x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -4.619047619x + x2 = 0 + -3
-4.619047619x + x2 = 0 + -3

Combine like terms: 0 + -3 = -3
-4.619047619x + x2 = -3

The x term is -4.619047619x.  Take half its coefficient (-2.30952381).
Square it (5.333900229) and add it to both sides.

Add '5.333900229' to each side of the equation.
-4.619047619x + 5.333900229 + x2 = -3 + 5.333900229

Reorder the terms:
5.333900229 + -4.619047619x + x2 = -3 + 5.333900229

Combine like terms: -3 + 5.333900229 = 2.333900229
5.333900229 + -4.619047619x + x2 = 2.333900229

Factor a perfect square on the left side:
(x + -2.30952381)(x + -2.30952381) = 2.333900229

Calculate the square root of the right side: 1.527710781

Break this problem into two subproblems by setting 
(x + -2.30952381) equal to 1.527710781 and -1.527710781.

Subproblem 1

x + -2.30952381 = 1.527710781 Simplifying x + -2.30952381 = 1.527710781 Reorder the terms: -2.30952381 + x = 1.527710781 Solving -2.30952381 + x = 1.527710781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.30952381' to each side of the equation. -2.30952381 + 2.30952381 + x = 1.527710781 + 2.30952381 Combine like terms: -2.30952381 + 2.30952381 = 0.00000000 0.00000000 + x = 1.527710781 + 2.30952381 x = 1.527710781 + 2.30952381 Combine like terms: 1.527710781 + 2.30952381 = 3.837234591 x = 3.837234591 Simplifying x = 3.837234591

Subproblem 2

x + -2.30952381 = -1.527710781 Simplifying x + -2.30952381 = -1.527710781 Reorder the terms: -2.30952381 + x = -1.527710781 Solving -2.30952381 + x = -1.527710781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.30952381' to each side of the equation. -2.30952381 + 2.30952381 + x = -1.527710781 + 2.30952381 Combine like terms: -2.30952381 + 2.30952381 = 0.00000000 0.00000000 + x = -1.527710781 + 2.30952381 x = -1.527710781 + 2.30952381 Combine like terms: -1.527710781 + 2.30952381 = 0.781813029 x = 0.781813029 Simplifying x = 0.781813029

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.837234591, 0.781813029}

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