2x^2-3x-21=0

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


Simplifying
2x2 + -3x + -21 = 0

Reorder the terms:
-21 + -3x + 2x2 = 0

Solving
-21 + -3x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-10.5 + -1.5x + x2 = 0

Move the constant term to the right:

Add '10.5' to each side of the equation.
-10.5 + -1.5x + 10.5 + x2 = 0 + 10.5

Reorder the terms:
-10.5 + 10.5 + -1.5x + x2 = 0 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + -1.5x + x2 = 0 + 10.5
-1.5x + x2 = 0 + 10.5

Combine like terms: 0 + 10.5 = 10.5
-1.5x + x2 = 10.5

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 10.5 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 10.5 + 0.5625

Combine like terms: 10.5 + 0.5625 = 11.0625
0.5625 + -1.5x + x2 = 11.0625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 11.0625

Calculate the square root of the right side: 3.326033674

Break this problem into two subproblems by setting 
(x + -0.75) equal to 3.326033674 and -3.326033674.

Subproblem 1

x + -0.75 = 3.326033674 Simplifying x + -0.75 = 3.326033674 Reorder the terms: -0.75 + x = 3.326033674 Solving -0.75 + x = 3.326033674 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 3.326033674 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 3.326033674 + 0.75 x = 3.326033674 + 0.75 Combine like terms: 3.326033674 + 0.75 = 4.076033674 x = 4.076033674 Simplifying x = 4.076033674

Subproblem 2

x + -0.75 = -3.326033674 Simplifying x + -0.75 = -3.326033674 Reorder the terms: -0.75 + x = -3.326033674 Solving -0.75 + x = -3.326033674 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -3.326033674 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -3.326033674 + 0.75 x = -3.326033674 + 0.75 Combine like terms: -3.326033674 + 0.75 = -2.576033674 x = -2.576033674 Simplifying x = -2.576033674

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.076033674, -2.576033674}

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