4x^2-61x-24=0

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Solution for 4x^2-61x-24=0 equation:


Simplifying
4x2 + -61x + -24 = 0

Reorder the terms:
-24 + -61x + 4x2 = 0

Solving
-24 + -61x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
-6 + -15.25x + x2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -15.25x + 6 + x2 = 0 + 6

Reorder the terms:
-6 + 6 + -15.25x + x2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -15.25x + x2 = 0 + 6
-15.25x + x2 = 0 + 6

Combine like terms: 0 + 6 = 6
-15.25x + x2 = 6

The x term is -15.25x.  Take half its coefficient (-7.625).
Square it (58.140625) and add it to both sides.

Add '58.140625' to each side of the equation.
-15.25x + 58.140625 + x2 = 6 + 58.140625

Reorder the terms:
58.140625 + -15.25x + x2 = 6 + 58.140625

Combine like terms: 6 + 58.140625 = 64.140625
58.140625 + -15.25x + x2 = 64.140625

Factor a perfect square on the left side:
(x + -7.625)(x + -7.625) = 64.140625

Calculate the square root of the right side: 8.00878424

Break this problem into two subproblems by setting 
(x + -7.625) equal to 8.00878424 and -8.00878424.

Subproblem 1

x + -7.625 = 8.00878424 Simplifying x + -7.625 = 8.00878424 Reorder the terms: -7.625 + x = 8.00878424 Solving -7.625 + x = 8.00878424 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.625' to each side of the equation. -7.625 + 7.625 + x = 8.00878424 + 7.625 Combine like terms: -7.625 + 7.625 = 0.000 0.000 + x = 8.00878424 + 7.625 x = 8.00878424 + 7.625 Combine like terms: 8.00878424 + 7.625 = 15.63378424 x = 15.63378424 Simplifying x = 15.63378424

Subproblem 2

x + -7.625 = -8.00878424 Simplifying x + -7.625 = -8.00878424 Reorder the terms: -7.625 + x = -8.00878424 Solving -7.625 + x = -8.00878424 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.625' to each side of the equation. -7.625 + 7.625 + x = -8.00878424 + 7.625 Combine like terms: -7.625 + 7.625 = 0.000 0.000 + x = -8.00878424 + 7.625 x = -8.00878424 + 7.625 Combine like terms: -8.00878424 + 7.625 = -0.38378424 x = -0.38378424 Simplifying x = -0.38378424

Solution

The solution to the problem is based on the solutions from the subproblems. x = {15.63378424, -0.38378424}

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