x(8x+77)=3x^2-7x+3

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Solution for x(8x+77)=3x^2-7x+3 equation:


Simplifying
x(8x + 77) = 3x2 + -7x + 3

Reorder the terms:
x(77 + 8x) = 3x2 + -7x + 3
(77 * x + 8x * x) = 3x2 + -7x + 3
(77x + 8x2) = 3x2 + -7x + 3

Reorder the terms:
77x + 8x2 = 3 + -7x + 3x2

Solving
77x + 8x2 = 3 + -7x + 3x2

Solving for variable 'x'.

Reorder the terms:
-3 + 77x + 7x + 8x2 + -3x2 = 3 + -7x + 3x2 + -3 + 7x + -3x2

Combine like terms: 77x + 7x = 84x
-3 + 84x + 8x2 + -3x2 = 3 + -7x + 3x2 + -3 + 7x + -3x2

Combine like terms: 8x2 + -3x2 = 5x2
-3 + 84x + 5x2 = 3 + -7x + 3x2 + -3 + 7x + -3x2

Reorder the terms:
-3 + 84x + 5x2 = 3 + -3 + -7x + 7x + 3x2 + -3x2

Combine like terms: 3 + -3 = 0
-3 + 84x + 5x2 = 0 + -7x + 7x + 3x2 + -3x2
-3 + 84x + 5x2 = -7x + 7x + 3x2 + -3x2

Combine like terms: -7x + 7x = 0
-3 + 84x + 5x2 = 0 + 3x2 + -3x2
-3 + 84x + 5x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
-3 + 84x + 5x2 = 0

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

Divide each side by '5'.
-0.6 + 16.8x + x2 = 0

Move the constant term to the right:

Add '0.6' to each side of the equation.
-0.6 + 16.8x + 0.6 + x2 = 0 + 0.6

Reorder the terms:
-0.6 + 0.6 + 16.8x + x2 = 0 + 0.6

Combine like terms: -0.6 + 0.6 = 0.0
0.0 + 16.8x + x2 = 0 + 0.6
16.8x + x2 = 0 + 0.6

Combine like terms: 0 + 0.6 = 0.6
16.8x + x2 = 0.6

The x term is 16.8x.  Take half its coefficient (8.4).
Square it (70.56) and add it to both sides.

Add '70.56' to each side of the equation.
16.8x + 70.56 + x2 = 0.6 + 70.56

Reorder the terms:
70.56 + 16.8x + x2 = 0.6 + 70.56

Combine like terms: 0.6 + 70.56 = 71.16
70.56 + 16.8x + x2 = 71.16

Factor a perfect square on the left side:
(x + 8.4)(x + 8.4) = 71.16

Calculate the square root of the right side: 8.435638684

Break this problem into two subproblems by setting 
(x + 8.4) equal to 8.435638684 and -8.435638684.

Subproblem 1

x + 8.4 = 8.435638684 Simplifying x + 8.4 = 8.435638684 Reorder the terms: 8.4 + x = 8.435638684 Solving 8.4 + x = 8.435638684 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.4' to each side of the equation. 8.4 + -8.4 + x = 8.435638684 + -8.4 Combine like terms: 8.4 + -8.4 = 0.0 0.0 + x = 8.435638684 + -8.4 x = 8.435638684 + -8.4 Combine like terms: 8.435638684 + -8.4 = 0.035638684 x = 0.035638684 Simplifying x = 0.035638684

Subproblem 2

x + 8.4 = -8.435638684 Simplifying x + 8.4 = -8.435638684 Reorder the terms: 8.4 + x = -8.435638684 Solving 8.4 + x = -8.435638684 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.4' to each side of the equation. 8.4 + -8.4 + x = -8.435638684 + -8.4 Combine like terms: 8.4 + -8.4 = 0.0 0.0 + x = -8.435638684 + -8.4 x = -8.435638684 + -8.4 Combine like terms: -8.435638684 + -8.4 = -16.835638684 x = -16.835638684 Simplifying x = -16.835638684

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.035638684, -16.835638684}

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