4x(78+49x)=77

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Solution for 4x(78+49x)=77 equation:


Simplifying
4x(78 + 49x) = 77
(78 * 4x + 49x * 4x) = 77
(312x + 196x2) = 77

Solving
312x + 196x2 = 77

Solving for variable 'x'.

Reorder the terms:
-77 + 312x + 196x2 = 77 + -77

Combine like terms: 77 + -77 = 0
-77 + 312x + 196x2 = 0

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

Divide each side by '196'.
-0.3928571429 + 1.591836735x + x2 = 0

Move the constant term to the right:

Add '0.3928571429' to each side of the equation.
-0.3928571429 + 1.591836735x + 0.3928571429 + x2 = 0 + 0.3928571429

Reorder the terms:
-0.3928571429 + 0.3928571429 + 1.591836735x + x2 = 0 + 0.3928571429

Combine like terms: -0.3928571429 + 0.3928571429 = 0.0000000000
0.0000000000 + 1.591836735x + x2 = 0 + 0.3928571429
1.591836735x + x2 = 0 + 0.3928571429

Combine like terms: 0 + 0.3928571429 = 0.3928571429
1.591836735x + x2 = 0.3928571429

The x term is 1.591836735x.  Take half its coefficient (0.7959183675).
Square it (0.6334860477) and add it to both sides.

Add '0.6334860477' to each side of the equation.
1.591836735x + 0.6334860477 + x2 = 0.3928571429 + 0.6334860477

Reorder the terms:
0.6334860477 + 1.591836735x + x2 = 0.3928571429 + 0.6334860477

Combine like terms: 0.3928571429 + 0.6334860477 = 1.0263431906
0.6334860477 + 1.591836735x + x2 = 1.0263431906

Factor a perfect square on the left side:
(x + 0.7959183675)(x + 0.7959183675) = 1.0263431906

Calculate the square root of the right side: 1.013085974

Break this problem into two subproblems by setting 
(x + 0.7959183675) equal to 1.013085974 and -1.013085974.

Subproblem 1

x + 0.7959183675 = 1.013085974 Simplifying x + 0.7959183675 = 1.013085974 Reorder the terms: 0.7959183675 + x = 1.013085974 Solving 0.7959183675 + x = 1.013085974 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7959183675' to each side of the equation. 0.7959183675 + -0.7959183675 + x = 1.013085974 + -0.7959183675 Combine like terms: 0.7959183675 + -0.7959183675 = 0.0000000000 0.0000000000 + x = 1.013085974 + -0.7959183675 x = 1.013085974 + -0.7959183675 Combine like terms: 1.013085974 + -0.7959183675 = 0.2171676065 x = 0.2171676065 Simplifying x = 0.2171676065

Subproblem 2

x + 0.7959183675 = -1.013085974 Simplifying x + 0.7959183675 = -1.013085974 Reorder the terms: 0.7959183675 + x = -1.013085974 Solving 0.7959183675 + x = -1.013085974 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7959183675' to each side of the equation. 0.7959183675 + -0.7959183675 + x = -1.013085974 + -0.7959183675 Combine like terms: 0.7959183675 + -0.7959183675 = 0.0000000000 0.0000000000 + x = -1.013085974 + -0.7959183675 x = -1.013085974 + -0.7959183675 Combine like terms: -1.013085974 + -0.7959183675 = -1.8090043415 x = -1.8090043415 Simplifying x = -1.8090043415

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.2171676065, -1.8090043415}

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