16x^2+78x+7=0

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Solution for 16x^2+78x+7=0 equation:


Simplifying
16x2 + 78x + 7 = 0

Reorder the terms:
7 + 78x + 16x2 = 0

Solving
7 + 78x + 16x2 = 0

Solving for variable 'x'.

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

Divide each side by '16'.
0.4375 + 4.875x + x2 = 0

Move the constant term to the right:

Add '-0.4375' to each side of the equation.
0.4375 + 4.875x + -0.4375 + x2 = 0 + -0.4375

Reorder the terms:
0.4375 + -0.4375 + 4.875x + x2 = 0 + -0.4375

Combine like terms: 0.4375 + -0.4375 = 0.0000
0.0000 + 4.875x + x2 = 0 + -0.4375
4.875x + x2 = 0 + -0.4375

Combine like terms: 0 + -0.4375 = -0.4375
4.875x + x2 = -0.4375

The x term is 4.875x.  Take half its coefficient (2.4375).
Square it (5.94140625) and add it to both sides.

Add '5.94140625' to each side of the equation.
4.875x + 5.94140625 + x2 = -0.4375 + 5.94140625

Reorder the terms:
5.94140625 + 4.875x + x2 = -0.4375 + 5.94140625

Combine like terms: -0.4375 + 5.94140625 = 5.50390625
5.94140625 + 4.875x + x2 = 5.50390625

Factor a perfect square on the left side:
(x + 2.4375)(x + 2.4375) = 5.50390625

Calculate the square root of the right side: 2.346040547

Break this problem into two subproblems by setting 
(x + 2.4375) equal to 2.346040547 and -2.346040547.

Subproblem 1

x + 2.4375 = 2.346040547 Simplifying x + 2.4375 = 2.346040547 Reorder the terms: 2.4375 + x = 2.346040547 Solving 2.4375 + x = 2.346040547 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.4375' to each side of the equation. 2.4375 + -2.4375 + x = 2.346040547 + -2.4375 Combine like terms: 2.4375 + -2.4375 = 0.0000 0.0000 + x = 2.346040547 + -2.4375 x = 2.346040547 + -2.4375 Combine like terms: 2.346040547 + -2.4375 = -0.091459453 x = -0.091459453 Simplifying x = -0.091459453

Subproblem 2

x + 2.4375 = -2.346040547 Simplifying x + 2.4375 = -2.346040547 Reorder the terms: 2.4375 + x = -2.346040547 Solving 2.4375 + x = -2.346040547 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.4375' to each side of the equation. 2.4375 + -2.4375 + x = -2.346040547 + -2.4375 Combine like terms: 2.4375 + -2.4375 = 0.0000 0.0000 + x = -2.346040547 + -2.4375 x = -2.346040547 + -2.4375 Combine like terms: -2.346040547 + -2.4375 = -4.783540547 x = -4.783540547 Simplifying x = -4.783540547

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.091459453, -4.783540547}

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