7x^2+11x-84=0

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Solution for 7x^2+11x-84=0 equation:


Simplifying
7x2 + 11x + -84 = 0

Reorder the terms:
-84 + 11x + 7x2 = 0

Solving
-84 + 11x + 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'.
-12 + 1.571428571x + x2 = 0

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + 1.571428571x + 12 + x2 = 0 + 12

Reorder the terms:
-12 + 12 + 1.571428571x + x2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + 1.571428571x + x2 = 0 + 12
1.571428571x + x2 = 0 + 12

Combine like terms: 0 + 12 = 12
1.571428571x + x2 = 12

The x term is 1.571428571x.  Take half its coefficient (0.7857142855).
Square it (0.6173469384) and add it to both sides.

Add '0.6173469384' to each side of the equation.
1.571428571x + 0.6173469384 + x2 = 12 + 0.6173469384

Reorder the terms:
0.6173469384 + 1.571428571x + x2 = 12 + 0.6173469384

Combine like terms: 12 + 0.6173469384 = 12.6173469384
0.6173469384 + 1.571428571x + x2 = 12.6173469384

Factor a perfect square on the left side:
(x + 0.7857142855)(x + 0.7857142855) = 12.6173469384

Calculate the square root of the right side: 3.552090503

Break this problem into two subproblems by setting 
(x + 0.7857142855) equal to 3.552090503 and -3.552090503.

Subproblem 1

x + 0.7857142855 = 3.552090503 Simplifying x + 0.7857142855 = 3.552090503 Reorder the terms: 0.7857142855 + x = 3.552090503 Solving 0.7857142855 + x = 3.552090503 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7857142855' to each side of the equation. 0.7857142855 + -0.7857142855 + x = 3.552090503 + -0.7857142855 Combine like terms: 0.7857142855 + -0.7857142855 = 0.0000000000 0.0000000000 + x = 3.552090503 + -0.7857142855 x = 3.552090503 + -0.7857142855 Combine like terms: 3.552090503 + -0.7857142855 = 2.7663762175 x = 2.7663762175 Simplifying x = 2.7663762175

Subproblem 2

x + 0.7857142855 = -3.552090503 Simplifying x + 0.7857142855 = -3.552090503 Reorder the terms: 0.7857142855 + x = -3.552090503 Solving 0.7857142855 + x = -3.552090503 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7857142855' to each side of the equation. 0.7857142855 + -0.7857142855 + x = -3.552090503 + -0.7857142855 Combine like terms: 0.7857142855 + -0.7857142855 = 0.0000000000 0.0000000000 + x = -3.552090503 + -0.7857142855 x = -3.552090503 + -0.7857142855 Combine like terms: -3.552090503 + -0.7857142855 = -4.3378047885 x = -4.3378047885 Simplifying x = -4.3378047885

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.7663762175, -4.3378047885}

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