7x^2+12x-21=0

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Solution for 7x^2+12x-21=0 equation:


Simplifying
7x2 + 12x + -21 = 0

Reorder the terms:
-21 + 12x + 7x2 = 0

Solving
-21 + 12x + 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 + 1.714285714x + x2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 1.714285714x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + 1.714285714x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 1.714285714x + x2 = 0 + 3
1.714285714x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
1.714285714x + x2 = 3

The x term is 1.714285714x.  Take half its coefficient (0.857142857).
Square it (0.7346938773) and add it to both sides.

Add '0.7346938773' to each side of the equation.
1.714285714x + 0.7346938773 + x2 = 3 + 0.7346938773

Reorder the terms:
0.7346938773 + 1.714285714x + x2 = 3 + 0.7346938773

Combine like terms: 3 + 0.7346938773 = 3.7346938773
0.7346938773 + 1.714285714x + x2 = 3.7346938773

Factor a perfect square on the left side:
(x + 0.857142857)(x + 0.857142857) = 3.7346938773

Calculate the square root of the right side: 1.932535608

Break this problem into two subproblems by setting 
(x + 0.857142857) equal to 1.932535608 and -1.932535608.

Subproblem 1

x + 0.857142857 = 1.932535608 Simplifying x + 0.857142857 = 1.932535608 Reorder the terms: 0.857142857 + x = 1.932535608 Solving 0.857142857 + x = 1.932535608 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + x = 1.932535608 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + x = 1.932535608 + -0.857142857 x = 1.932535608 + -0.857142857 Combine like terms: 1.932535608 + -0.857142857 = 1.075392751 x = 1.075392751 Simplifying x = 1.075392751

Subproblem 2

x + 0.857142857 = -1.932535608 Simplifying x + 0.857142857 = -1.932535608 Reorder the terms: 0.857142857 + x = -1.932535608 Solving 0.857142857 + x = -1.932535608 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + x = -1.932535608 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + x = -1.932535608 + -0.857142857 x = -1.932535608 + -0.857142857 Combine like terms: -1.932535608 + -0.857142857 = -2.789678465 x = -2.789678465 Simplifying x = -2.789678465

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.075392751, -2.789678465}

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