4x^2+11x+5=0

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Solution for 4x^2+11x+5=0 equation:


Simplifying
4x2 + 11x + 5 = 0

Reorder the terms:
5 + 11x + 4x2 = 0

Solving
5 + 11x + 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'.
1.25 + 2.75x + x2 = 0

Move the constant term to the right:

Add '-1.25' to each side of the equation.
1.25 + 2.75x + -1.25 + x2 = 0 + -1.25

Reorder the terms:
1.25 + -1.25 + 2.75x + x2 = 0 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + 2.75x + x2 = 0 + -1.25
2.75x + x2 = 0 + -1.25

Combine like terms: 0 + -1.25 = -1.25
2.75x + x2 = -1.25

The x term is 2.75x.  Take half its coefficient (1.375).
Square it (1.890625) and add it to both sides.

Add '1.890625' to each side of the equation.
2.75x + 1.890625 + x2 = -1.25 + 1.890625

Reorder the terms:
1.890625 + 2.75x + x2 = -1.25 + 1.890625

Combine like terms: -1.25 + 1.890625 = 0.640625
1.890625 + 2.75x + x2 = 0.640625

Factor a perfect square on the left side:
(x + 1.375)(x + 1.375) = 0.640625

Calculate the square root of the right side: 0.80039053

Break this problem into two subproblems by setting 
(x + 1.375) equal to 0.80039053 and -0.80039053.

Subproblem 1

x + 1.375 = 0.80039053 Simplifying x + 1.375 = 0.80039053 Reorder the terms: 1.375 + x = 0.80039053 Solving 1.375 + x = 0.80039053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + x = 0.80039053 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + x = 0.80039053 + -1.375 x = 0.80039053 + -1.375 Combine like terms: 0.80039053 + -1.375 = -0.57460947 x = -0.57460947 Simplifying x = -0.57460947

Subproblem 2

x + 1.375 = -0.80039053 Simplifying x + 1.375 = -0.80039053 Reorder the terms: 1.375 + x = -0.80039053 Solving 1.375 + x = -0.80039053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + x = -0.80039053 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + x = -0.80039053 + -1.375 x = -0.80039053 + -1.375 Combine like terms: -0.80039053 + -1.375 = -2.17539053 x = -2.17539053 Simplifying x = -2.17539053

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.57460947, -2.17539053}

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