4x^2+21x+24=0

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Solution for 4x^2+21x+24=0 equation:


Simplifying
4x2 + 21x + 24 = 0

Reorder the terms:
24 + 21x + 4x2 = 0

Solving
24 + 21x + 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 + 5.25x + x2 = 0

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + 5.25x + -6 + x2 = 0 + -6

Reorder the terms:
6 + -6 + 5.25x + x2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + 5.25x + x2 = 0 + -6
5.25x + x2 = 0 + -6

Combine like terms: 0 + -6 = -6
5.25x + x2 = -6

The x term is 5.25x.  Take half its coefficient (2.625).
Square it (6.890625) and add it to both sides.

Add '6.890625' to each side of the equation.
5.25x + 6.890625 + x2 = -6 + 6.890625

Reorder the terms:
6.890625 + 5.25x + x2 = -6 + 6.890625

Combine like terms: -6 + 6.890625 = 0.890625
6.890625 + 5.25x + x2 = 0.890625

Factor a perfect square on the left side:
(x + 2.625)(x + 2.625) = 0.890625

Calculate the square root of the right side: 0.943729304

Break this problem into two subproblems by setting 
(x + 2.625) equal to 0.943729304 and -0.943729304.

Subproblem 1

x + 2.625 = 0.943729304 Simplifying x + 2.625 = 0.943729304 Reorder the terms: 2.625 + x = 0.943729304 Solving 2.625 + x = 0.943729304 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.625' to each side of the equation. 2.625 + -2.625 + x = 0.943729304 + -2.625 Combine like terms: 2.625 + -2.625 = 0.000 0.000 + x = 0.943729304 + -2.625 x = 0.943729304 + -2.625 Combine like terms: 0.943729304 + -2.625 = -1.681270696 x = -1.681270696 Simplifying x = -1.681270696

Subproblem 2

x + 2.625 = -0.943729304 Simplifying x + 2.625 = -0.943729304 Reorder the terms: 2.625 + x = -0.943729304 Solving 2.625 + x = -0.943729304 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.625' to each side of the equation. 2.625 + -2.625 + x = -0.943729304 + -2.625 Combine like terms: 2.625 + -2.625 = 0.000 0.000 + x = -0.943729304 + -2.625 x = -0.943729304 + -2.625 Combine like terms: -0.943729304 + -2.625 = -3.568729304 x = -3.568729304 Simplifying x = -3.568729304

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.681270696, -3.568729304}

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