4x^2-24=-x

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


Simplifying
4x2 + -24 = -1x

Reorder the terms:
-24 + 4x2 = -1x

Solving
-24 + 4x2 = -1x

Solving for variable 'x'.

Reorder the terms:
-24 + x + 4x2 = -1x + x

Combine like terms: -1x + x = 0
-24 + x + 4x2 = 0

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

Divide each side by '4'.
-6 + 0.25x + x2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 0.25x + 6 + x2 = 0 + 6

Reorder the terms:
-6 + 6 + 0.25x + x2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 0.25x + x2 = 0 + 6
0.25x + x2 = 0 + 6

Combine like terms: 0 + 6 = 6
0.25x + x2 = 6

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.25x + 0.25 + x2 = 6 + 0.25

Reorder the terms:
0.25 + 0.25x + x2 = 6 + 0.25

Combine like terms: 6 + 0.25 = 6.25
0.25 + 0.25x + x2 = 6.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 6.25

Calculate the square root of the right side: 2.5

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.5 and -2.5.

Subproblem 1

x + 0.5 = 2.5 Simplifying x + 0.5 = 2.5 Reorder the terms: 0.5 + x = 2.5 Solving 0.5 + x = 2.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 2.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 2.5 + -0.5 x = 2.5 + -0.5 Combine like terms: 2.5 + -0.5 = 2 x = 2 Simplifying x = 2

Subproblem 2

x + 0.5 = -2.5 Simplifying x + 0.5 = -2.5 Reorder the terms: 0.5 + x = -2.5 Solving 0.5 + x = -2.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -2.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -2.5 + -0.5 x = -2.5 + -0.5 Combine like terms: -2.5 + -0.5 = -3 x = -3 Simplifying x = -3

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2, -3}

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