9x^2+15x-20=0

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Solution for 9x^2+15x-20=0 equation:


Simplifying
9x2 + 15x + -20 = 0

Reorder the terms:
-20 + 15x + 9x2 = 0

Solving
-20 + 15x + 9x2 = 0

Solving for variable 'x'.

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

Divide each side by '9'.
-2.222222222 + 1.666666667x + x2 = 0

Move the constant term to the right:

Add '2.222222222' to each side of the equation.
-2.222222222 + 1.666666667x + 2.222222222 + x2 = 0 + 2.222222222

Reorder the terms:
-2.222222222 + 2.222222222 + 1.666666667x + x2 = 0 + 2.222222222

Combine like terms: -2.222222222 + 2.222222222 = 0.000000000
0.000000000 + 1.666666667x + x2 = 0 + 2.222222222
1.666666667x + x2 = 0 + 2.222222222

Combine like terms: 0 + 2.222222222 = 2.222222222
1.666666667x + x2 = 2.222222222

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 = 2.222222222 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667x + x2 = 2.222222222 + 0.6944444447

Combine like terms: 2.222222222 + 0.6944444447 = 2.9166666667
0.6944444447 + 1.666666667x + x2 = 2.9166666667

Factor a perfect square on the left side:
(x + 0.8333333335)(x + 0.8333333335) = 2.9166666667

Calculate the square root of the right side: 1.707825128

Break this problem into two subproblems by setting 
(x + 0.8333333335) equal to 1.707825128 and -1.707825128.

Subproblem 1

x + 0.8333333335 = 1.707825128 Simplifying x + 0.8333333335 = 1.707825128 Reorder the terms: 0.8333333335 + x = 1.707825128 Solving 0.8333333335 + x = 1.707825128 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 = 1.707825128 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = 1.707825128 + -0.8333333335 x = 1.707825128 + -0.8333333335 Combine like terms: 1.707825128 + -0.8333333335 = 0.8744917945 x = 0.8744917945 Simplifying x = 0.8744917945

Subproblem 2

x + 0.8333333335 = -1.707825128 Simplifying x + 0.8333333335 = -1.707825128 Reorder the terms: 0.8333333335 + x = -1.707825128 Solving 0.8333333335 + x = -1.707825128 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 = -1.707825128 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = -1.707825128 + -0.8333333335 x = -1.707825128 + -0.8333333335 Combine like terms: -1.707825128 + -0.8333333335 = -2.5411584615 x = -2.5411584615 Simplifying x = -2.5411584615

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.8744917945, -2.5411584615}

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