19x^2-30x-77=0

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Solution for 19x^2-30x-77=0 equation:


Simplifying
19x2 + -30x + -77 = 0

Reorder the terms:
-77 + -30x + 19x2 = 0

Solving
-77 + -30x + 19x2 = 0

Solving for variable 'x'.

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

Divide each side by '19'.
-4.052631579 + -1.578947368x + x2 = 0

Move the constant term to the right:

Add '4.052631579' to each side of the equation.
-4.052631579 + -1.578947368x + 4.052631579 + x2 = 0 + 4.052631579

Reorder the terms:
-4.052631579 + 4.052631579 + -1.578947368x + x2 = 0 + 4.052631579

Combine like terms: -4.052631579 + 4.052631579 = 0.000000000
0.000000000 + -1.578947368x + x2 = 0 + 4.052631579
-1.578947368x + x2 = 0 + 4.052631579

Combine like terms: 0 + 4.052631579 = 4.052631579
-1.578947368x + x2 = 4.052631579

The x term is -1.578947368x.  Take half its coefficient (-0.789473684).
Square it (0.6232686977) and add it to both sides.

Add '0.6232686977' to each side of the equation.
-1.578947368x + 0.6232686977 + x2 = 4.052631579 + 0.6232686977

Reorder the terms:
0.6232686977 + -1.578947368x + x2 = 4.052631579 + 0.6232686977

Combine like terms: 4.052631579 + 0.6232686977 = 4.6759002767
0.6232686977 + -1.578947368x + x2 = 4.6759002767

Factor a perfect square on the left side:
(x + -0.789473684)(x + -0.789473684) = 4.6759002767

Calculate the square root of the right side: 2.162383009

Break this problem into two subproblems by setting 
(x + -0.789473684) equal to 2.162383009 and -2.162383009.

Subproblem 1

x + -0.789473684 = 2.162383009 Simplifying x + -0.789473684 = 2.162383009 Reorder the terms: -0.789473684 + x = 2.162383009 Solving -0.789473684 + x = 2.162383009 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.789473684' to each side of the equation. -0.789473684 + 0.789473684 + x = 2.162383009 + 0.789473684 Combine like terms: -0.789473684 + 0.789473684 = 0.000000000 0.000000000 + x = 2.162383009 + 0.789473684 x = 2.162383009 + 0.789473684 Combine like terms: 2.162383009 + 0.789473684 = 2.951856693 x = 2.951856693 Simplifying x = 2.951856693

Subproblem 2

x + -0.789473684 = -2.162383009 Simplifying x + -0.789473684 = -2.162383009 Reorder the terms: -0.789473684 + x = -2.162383009 Solving -0.789473684 + x = -2.162383009 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.789473684' to each side of the equation. -0.789473684 + 0.789473684 + x = -2.162383009 + 0.789473684 Combine like terms: -0.789473684 + 0.789473684 = 0.000000000 0.000000000 + x = -2.162383009 + 0.789473684 x = -2.162383009 + 0.789473684 Combine like terms: -2.162383009 + 0.789473684 = -1.372909325 x = -1.372909325 Simplifying x = -1.372909325

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.951856693, -1.372909325}

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