4x^2=-10x-5

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Solution for 4x^2=-10x-5 equation:


Simplifying
4x2 = -10x + -5

Reorder the terms:
4x2 = -5 + -10x

Solving
4x2 = -5 + -10x

Solving for variable 'x'.

Reorder the terms:
5 + 10x + 4x2 = -5 + -10x + 5 + 10x

Reorder the terms:
5 + 10x + 4x2 = -5 + 5 + -10x + 10x

Combine like terms: -5 + 5 = 0
5 + 10x + 4x2 = 0 + -10x + 10x
5 + 10x + 4x2 = -10x + 10x

Combine like terms: -10x + 10x = 0
5 + 10x + 4x2 = 0

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

Divide each side by '4'.
1.25 + 2.5x + x2 = 0

Move the constant term to the right:

Add '-1.25' to each side of the equation.
1.25 + 2.5x + -1.25 + x2 = 0 + -1.25

Reorder the terms:
1.25 + -1.25 + 2.5x + x2 = 0 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + 2.5x + x2 = 0 + -1.25
2.5x + x2 = 0 + -1.25

Combine like terms: 0 + -1.25 = -1.25
2.5x + x2 = -1.25

The x term is 2.5x.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5x + 1.5625 + x2 = -1.25 + 1.5625

Reorder the terms:
1.5625 + 2.5x + x2 = -1.25 + 1.5625

Combine like terms: -1.25 + 1.5625 = 0.3125
1.5625 + 2.5x + x2 = 0.3125

Factor a perfect square on the left side:
(x + 1.25)(x + 1.25) = 0.3125

Calculate the square root of the right side: 0.559016994

Break this problem into two subproblems by setting 
(x + 1.25) equal to 0.559016994 and -0.559016994.

Subproblem 1

x + 1.25 = 0.559016994 Simplifying x + 1.25 = 0.559016994 Reorder the terms: 1.25 + x = 0.559016994 Solving 1.25 + x = 0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = 0.559016994 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = 0.559016994 + -1.25 x = 0.559016994 + -1.25 Combine like terms: 0.559016994 + -1.25 = -0.690983006 x = -0.690983006 Simplifying x = -0.690983006

Subproblem 2

x + 1.25 = -0.559016994 Simplifying x + 1.25 = -0.559016994 Reorder the terms: 1.25 + x = -0.559016994 Solving 1.25 + x = -0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = -0.559016994 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = -0.559016994 + -1.25 x = -0.559016994 + -1.25 Combine like terms: -0.559016994 + -1.25 = -1.809016994 x = -1.809016994 Simplifying x = -1.809016994

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.690983006, -1.809016994}

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