x(2x+5)=420

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Solution for x(2x+5)=420 equation:


Simplifying
x(2x + 5) = 420

Reorder the terms:
x(5 + 2x) = 420
(5 * x + 2x * x) = 420
(5x + 2x2) = 420

Solving
5x + 2x2 = 420

Solving for variable 'x'.

Reorder the terms:
-420 + 5x + 2x2 = 420 + -420

Combine like terms: 420 + -420 = 0
-420 + 5x + 2x2 = 0

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

Divide each side by '2'.
-210 + 2.5x + x2 = 0

Move the constant term to the right:

Add '210' to each side of the equation.
-210 + 2.5x + 210 + x2 = 0 + 210

Reorder the terms:
-210 + 210 + 2.5x + x2 = 0 + 210

Combine like terms: -210 + 210 = 0
0 + 2.5x + x2 = 0 + 210
2.5x + x2 = 0 + 210

Combine like terms: 0 + 210 = 210
2.5x + x2 = 210

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 = 210 + 1.5625

Reorder the terms:
1.5625 + 2.5x + x2 = 210 + 1.5625

Combine like terms: 210 + 1.5625 = 211.5625
1.5625 + 2.5x + x2 = 211.5625

Factor a perfect square on the left side:
(x + 1.25)(x + 1.25) = 211.5625

Calculate the square root of the right side: 14.545188208

Break this problem into two subproblems by setting 
(x + 1.25) equal to 14.545188208 and -14.545188208.

Subproblem 1

x + 1.25 = 14.545188208 Simplifying x + 1.25 = 14.545188208 Reorder the terms: 1.25 + x = 14.545188208 Solving 1.25 + x = 14.545188208 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 = 14.545188208 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = 14.545188208 + -1.25 x = 14.545188208 + -1.25 Combine like terms: 14.545188208 + -1.25 = 13.295188208 x = 13.295188208 Simplifying x = 13.295188208

Subproblem 2

x + 1.25 = -14.545188208 Simplifying x + 1.25 = -14.545188208 Reorder the terms: 1.25 + x = -14.545188208 Solving 1.25 + x = -14.545188208 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 = -14.545188208 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = -14.545188208 + -1.25 x = -14.545188208 + -1.25 Combine like terms: -14.545188208 + -1.25 = -15.795188208 x = -15.795188208 Simplifying x = -15.795188208

Solution

The solution to the problem is based on the solutions from the subproblems. x = {13.295188208, -15.795188208}

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