(7x+5)(x+4)=8

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Solution for (7x+5)(x+4)=8 equation:


Simplifying
(7x + 5)(x + 4) = 8

Reorder the terms:
(5 + 7x)(x + 4) = 8

Reorder the terms:
(5 + 7x)(4 + x) = 8

Multiply (5 + 7x) * (4 + x)
(5(4 + x) + 7x * (4 + x)) = 8
((4 * 5 + x * 5) + 7x * (4 + x)) = 8
((20 + 5x) + 7x * (4 + x)) = 8
(20 + 5x + (4 * 7x + x * 7x)) = 8
(20 + 5x + (28x + 7x2)) = 8

Combine like terms: 5x + 28x = 33x
(20 + 33x + 7x2) = 8

Solving
20 + 33x + 7x2 = 8

Solving for variable 'x'.

Reorder the terms:
20 + -8 + 33x + 7x2 = 8 + -8

Combine like terms: 20 + -8 = 12
12 + 33x + 7x2 = 8 + -8

Combine like terms: 8 + -8 = 0
12 + 33x + 7x2 = 0

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

Divide each side by '7'.
1.714285714 + 4.714285714x + x2 = 0

Move the constant term to the right:

Add '-1.714285714' to each side of the equation.
1.714285714 + 4.714285714x + -1.714285714 + x2 = 0 + -1.714285714

Reorder the terms:
1.714285714 + -1.714285714 + 4.714285714x + x2 = 0 + -1.714285714

Combine like terms: 1.714285714 + -1.714285714 = 0.000000000
0.000000000 + 4.714285714x + x2 = 0 + -1.714285714
4.714285714x + x2 = 0 + -1.714285714

Combine like terms: 0 + -1.714285714 = -1.714285714
4.714285714x + x2 = -1.714285714

The x term is 4.714285714x.  Take half its coefficient (2.357142857).
Square it (5.556122448) and add it to both sides.

Add '5.556122448' to each side of the equation.
4.714285714x + 5.556122448 + x2 = -1.714285714 + 5.556122448

Reorder the terms:
5.556122448 + 4.714285714x + x2 = -1.714285714 + 5.556122448

Combine like terms: -1.714285714 + 5.556122448 = 3.841836734
5.556122448 + 4.714285714x + x2 = 3.841836734

Factor a perfect square on the left side:
(x + 2.357142857)(x + 2.357142857) = 3.841836734

Calculate the square root of the right side: 1.96006039

Break this problem into two subproblems by setting 
(x + 2.357142857) equal to 1.96006039 and -1.96006039.

Subproblem 1

x + 2.357142857 = 1.96006039 Simplifying x + 2.357142857 = 1.96006039 Reorder the terms: 2.357142857 + x = 1.96006039 Solving 2.357142857 + x = 1.96006039 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.357142857' to each side of the equation. 2.357142857 + -2.357142857 + x = 1.96006039 + -2.357142857 Combine like terms: 2.357142857 + -2.357142857 = 0.000000000 0.000000000 + x = 1.96006039 + -2.357142857 x = 1.96006039 + -2.357142857 Combine like terms: 1.96006039 + -2.357142857 = -0.397082467 x = -0.397082467 Simplifying x = -0.397082467

Subproblem 2

x + 2.357142857 = -1.96006039 Simplifying x + 2.357142857 = -1.96006039 Reorder the terms: 2.357142857 + x = -1.96006039 Solving 2.357142857 + x = -1.96006039 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.357142857' to each side of the equation. 2.357142857 + -2.357142857 + x = -1.96006039 + -2.357142857 Combine like terms: 2.357142857 + -2.357142857 = 0.000000000 0.000000000 + x = -1.96006039 + -2.357142857 x = -1.96006039 + -2.357142857 Combine like terms: -1.96006039 + -2.357142857 = -4.317203247 x = -4.317203247 Simplifying x = -4.317203247

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.397082467, -4.317203247}

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