15x^2-19x-24=0

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Solution for 15x^2-19x-24=0 equation:


Simplifying
15x2 + -19x + -24 = 0

Reorder the terms:
-24 + -19x + 15x2 = 0

Solving
-24 + -19x + 15x2 = 0

Solving for variable 'x'.

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

Divide each side by '15'.
-1.6 + -1.266666667x + x2 = 0

Move the constant term to the right:

Add '1.6' to each side of the equation.
-1.6 + -1.266666667x + 1.6 + x2 = 0 + 1.6

Reorder the terms:
-1.6 + 1.6 + -1.266666667x + x2 = 0 + 1.6

Combine like terms: -1.6 + 1.6 = 0.0
0.0 + -1.266666667x + x2 = 0 + 1.6
-1.266666667x + x2 = 0 + 1.6

Combine like terms: 0 + 1.6 = 1.6
-1.266666667x + x2 = 1.6

The x term is -1.266666667x.  Take half its coefficient (-0.6333333335).
Square it (0.4011111113) and add it to both sides.

Add '0.4011111113' to each side of the equation.
-1.266666667x + 0.4011111113 + x2 = 1.6 + 0.4011111113

Reorder the terms:
0.4011111113 + -1.266666667x + x2 = 1.6 + 0.4011111113

Combine like terms: 1.6 + 0.4011111113 = 2.0011111113
0.4011111113 + -1.266666667x + x2 = 2.0011111113

Factor a perfect square on the left side:
(x + -0.6333333335)(x + -0.6333333335) = 2.0011111113

Calculate the square root of the right side: 1.414606345

Break this problem into two subproblems by setting 
(x + -0.6333333335) equal to 1.414606345 and -1.414606345.

Subproblem 1

x + -0.6333333335 = 1.414606345 Simplifying x + -0.6333333335 = 1.414606345 Reorder the terms: -0.6333333335 + x = 1.414606345 Solving -0.6333333335 + x = 1.414606345 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6333333335' to each side of the equation. -0.6333333335 + 0.6333333335 + x = 1.414606345 + 0.6333333335 Combine like terms: -0.6333333335 + 0.6333333335 = 0.0000000000 0.0000000000 + x = 1.414606345 + 0.6333333335 x = 1.414606345 + 0.6333333335 Combine like terms: 1.414606345 + 0.6333333335 = 2.0479396785 x = 2.0479396785 Simplifying x = 2.0479396785

Subproblem 2

x + -0.6333333335 = -1.414606345 Simplifying x + -0.6333333335 = -1.414606345 Reorder the terms: -0.6333333335 + x = -1.414606345 Solving -0.6333333335 + x = -1.414606345 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6333333335' to each side of the equation. -0.6333333335 + 0.6333333335 + x = -1.414606345 + 0.6333333335 Combine like terms: -0.6333333335 + 0.6333333335 = 0.0000000000 0.0000000000 + x = -1.414606345 + 0.6333333335 x = -1.414606345 + 0.6333333335 Combine like terms: -1.414606345 + 0.6333333335 = -0.7812730115 x = -0.7812730115 Simplifying x = -0.7812730115

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.0479396785, -0.7812730115}

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