3x^2-5x-21=0

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Solution for 3x^2-5x-21=0 equation:


Simplifying
3x2 + -5x + -21 = 0

Reorder the terms:
-21 + -5x + 3x2 = 0

Solving
-21 + -5x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
-7 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -1.666666667x + 7 + x2 = 0 + 7

Reorder the terms:
-7 + 7 + -1.666666667x + x2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -1.666666667x + x2 = 0 + 7
-1.666666667x + x2 = 0 + 7

Combine like terms: 0 + 7 = 7
-1.666666667x + x2 = 7

The x term is -1.666666667x.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667x + 0.6944444447 + x2 = 7 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = 7 + 0.6944444447

Combine like terms: 7 + 0.6944444447 = 7.6944444447
0.6944444447 + -1.666666667x + x2 = 7.6944444447

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 7.6944444447

Calculate the square root of the right side: 2.773886163

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 2.773886163 and -2.773886163.

Subproblem 1

x + -0.8333333335 = 2.773886163 Simplifying x + -0.8333333335 = 2.773886163 Reorder the terms: -0.8333333335 + x = 2.773886163 Solving -0.8333333335 + x = 2.773886163 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = 2.773886163 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 2.773886163 + 0.8333333335 x = 2.773886163 + 0.8333333335 Combine like terms: 2.773886163 + 0.8333333335 = 3.6072194965 x = 3.6072194965 Simplifying x = 3.6072194965

Subproblem 2

x + -0.8333333335 = -2.773886163 Simplifying x + -0.8333333335 = -2.773886163 Reorder the terms: -0.8333333335 + x = -2.773886163 Solving -0.8333333335 + x = -2.773886163 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = -2.773886163 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -2.773886163 + 0.8333333335 x = -2.773886163 + 0.8333333335 Combine like terms: -2.773886163 + 0.8333333335 = -1.9405528295 x = -1.9405528295 Simplifying x = -1.9405528295

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.6072194965, -1.9405528295}

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