4x^2+20x-415=0

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Solution for 4x^2+20x-415=0 equation:


Simplifying
4x2 + 20x + -415 = 0

Reorder the terms:
-415 + 20x + 4x2 = 0

Solving
-415 + 20x + 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'.
-103.75 + 5x + x2 = 0

Move the constant term to the right:

Add '103.75' to each side of the equation.
-103.75 + 5x + 103.75 + x2 = 0 + 103.75

Reorder the terms:
-103.75 + 103.75 + 5x + x2 = 0 + 103.75

Combine like terms: -103.75 + 103.75 = 0.00
0.00 + 5x + x2 = 0 + 103.75
5x + x2 = 0 + 103.75

Combine like terms: 0 + 103.75 = 103.75
5x + x2 = 103.75

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = 103.75 + 6.25

Reorder the terms:
6.25 + 5x + x2 = 103.75 + 6.25

Combine like terms: 103.75 + 6.25 = 110
6.25 + 5x + x2 = 110

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 110

Calculate the square root of the right side: 10.488088482

Break this problem into two subproblems by setting 
(x + 2.5) equal to 10.488088482 and -10.488088482.

Subproblem 1

x + 2.5 = 10.488088482 Simplifying x + 2.5 = 10.488088482 Reorder the terms: 2.5 + x = 10.488088482 Solving 2.5 + x = 10.488088482 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 10.488088482 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 10.488088482 + -2.5 x = 10.488088482 + -2.5 Combine like terms: 10.488088482 + -2.5 = 7.988088482 x = 7.988088482 Simplifying x = 7.988088482

Subproblem 2

x + 2.5 = -10.488088482 Simplifying x + 2.5 = -10.488088482 Reorder the terms: 2.5 + x = -10.488088482 Solving 2.5 + x = -10.488088482 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -10.488088482 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -10.488088482 + -2.5 x = -10.488088482 + -2.5 Combine like terms: -10.488088482 + -2.5 = -12.988088482 x = -12.988088482 Simplifying x = -12.988088482

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.988088482, -12.988088482}

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