7x^2-31x+21=0

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Solution for 7x^2-31x+21=0 equation:


Simplifying
7x2 + -31x + 21 = 0

Reorder the terms:
21 + -31x + 7x2 = 0

Solving
21 + -31x + 7x2 = 0

Solving for variable 'x'.

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

Divide each side by '7'.
3 + -4.428571429x + x2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -4.428571429x + -3 + x2 = 0 + -3

Reorder the terms:
3 + -3 + -4.428571429x + x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -4.428571429x + x2 = 0 + -3
-4.428571429x + x2 = 0 + -3

Combine like terms: 0 + -3 = -3
-4.428571429x + x2 = -3

The x term is -4.428571429x.  Take half its coefficient (-2.214285715).
Square it (4.903061228) and add it to both sides.

Add '4.903061228' to each side of the equation.
-4.428571429x + 4.903061228 + x2 = -3 + 4.903061228

Reorder the terms:
4.903061228 + -4.428571429x + x2 = -3 + 4.903061228

Combine like terms: -3 + 4.903061228 = 1.903061228
4.903061228 + -4.428571429x + x2 = 1.903061228

Factor a perfect square on the left side:
(x + -2.214285715)(x + -2.214285715) = 1.903061228

Calculate the square root of the right side: 1.379514852

Break this problem into two subproblems by setting 
(x + -2.214285715) equal to 1.379514852 and -1.379514852.

Subproblem 1

x + -2.214285715 = 1.379514852 Simplifying x + -2.214285715 = 1.379514852 Reorder the terms: -2.214285715 + x = 1.379514852 Solving -2.214285715 + x = 1.379514852 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.214285715' to each side of the equation. -2.214285715 + 2.214285715 + x = 1.379514852 + 2.214285715 Combine like terms: -2.214285715 + 2.214285715 = 0.000000000 0.000000000 + x = 1.379514852 + 2.214285715 x = 1.379514852 + 2.214285715 Combine like terms: 1.379514852 + 2.214285715 = 3.593800567 x = 3.593800567 Simplifying x = 3.593800567

Subproblem 2

x + -2.214285715 = -1.379514852 Simplifying x + -2.214285715 = -1.379514852 Reorder the terms: -2.214285715 + x = -1.379514852 Solving -2.214285715 + x = -1.379514852 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.214285715' to each side of the equation. -2.214285715 + 2.214285715 + x = -1.379514852 + 2.214285715 Combine like terms: -2.214285715 + 2.214285715 = 0.000000000 0.000000000 + x = -1.379514852 + 2.214285715 x = -1.379514852 + 2.214285715 Combine like terms: -1.379514852 + 2.214285715 = 0.834770863 x = 0.834770863 Simplifying x = 0.834770863

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.593800567, 0.834770863}

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