21x^2+14x-5=0

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Solution for 21x^2+14x-5=0 equation:


Simplifying
21x2 + 14x + -5 = 0

Reorder the terms:
-5 + 14x + 21x2 = 0

Solving
-5 + 14x + 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'.
-0.2380952381 + 0.6666666667x + x2 = 0

Move the constant term to the right:

Add '0.2380952381' to each side of the equation.
-0.2380952381 + 0.6666666667x + 0.2380952381 + x2 = 0 + 0.2380952381

Reorder the terms:
-0.2380952381 + 0.2380952381 + 0.6666666667x + x2 = 0 + 0.2380952381

Combine like terms: -0.2380952381 + 0.2380952381 = 0.0000000000
0.0000000000 + 0.6666666667x + x2 = 0 + 0.2380952381
0.6666666667x + x2 = 0 + 0.2380952381

Combine like terms: 0 + 0.2380952381 = 0.2380952381
0.6666666667x + x2 = 0.2380952381

The x term is 0.6666666667x.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667x + 0.1111111112 + x2 = 0.2380952381 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 0.2380952381 + 0.1111111112

Combine like terms: 0.2380952381 + 0.1111111112 = 0.3492063493
0.1111111112 + 0.6666666667x + x2 = 0.3492063493

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 0.3492063493

Calculate the square root of the right side: 0.59093684

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 0.59093684 and -0.59093684.

Subproblem 1

x + 0.3333333334 = 0.59093684 Simplifying x + 0.3333333334 = 0.59093684 Reorder the terms: 0.3333333334 + x = 0.59093684 Solving 0.3333333334 + x = 0.59093684 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = 0.59093684 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 0.59093684 + -0.3333333334 x = 0.59093684 + -0.3333333334 Combine like terms: 0.59093684 + -0.3333333334 = 0.2576035066 x = 0.2576035066 Simplifying x = 0.2576035066

Subproblem 2

x + 0.3333333334 = -0.59093684 Simplifying x + 0.3333333334 = -0.59093684 Reorder the terms: 0.3333333334 + x = -0.59093684 Solving 0.3333333334 + x = -0.59093684 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = -0.59093684 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -0.59093684 + -0.3333333334 x = -0.59093684 + -0.3333333334 Combine like terms: -0.59093684 + -0.3333333334 = -0.9242701734 x = -0.9242701734 Simplifying x = -0.9242701734

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.2576035066, -0.9242701734}

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