(4t+5)(7t+5)=-3t

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Solution for (4t+5)(7t+5)=-3t equation:


Simplifying
(4t + 5)(7t + 5) = -3t

Reorder the terms:
(5 + 4t)(7t + 5) = -3t

Reorder the terms:
(5 + 4t)(5 + 7t) = -3t

Multiply (5 + 4t) * (5 + 7t)
(5(5 + 7t) + 4t * (5 + 7t)) = -3t
((5 * 5 + 7t * 5) + 4t * (5 + 7t)) = -3t
((25 + 35t) + 4t * (5 + 7t)) = -3t
(25 + 35t + (5 * 4t + 7t * 4t)) = -3t
(25 + 35t + (20t + 28t2)) = -3t

Combine like terms: 35t + 20t = 55t
(25 + 55t + 28t2) = -3t

Solving
25 + 55t + 28t2 = -3t

Solving for variable 't'.

Reorder the terms:
25 + 55t + 3t + 28t2 = -3t + 3t

Combine like terms: 55t + 3t = 58t
25 + 58t + 28t2 = -3t + 3t

Combine like terms: -3t + 3t = 0
25 + 58t + 28t2 = 0

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

Divide each side by '28'.
0.8928571429 + 2.071428571t + t2 = 0

Move the constant term to the right:

Add '-0.8928571429' to each side of the equation.
0.8928571429 + 2.071428571t + -0.8928571429 + t2 = 0 + -0.8928571429

Reorder the terms:
0.8928571429 + -0.8928571429 + 2.071428571t + t2 = 0 + -0.8928571429

Combine like terms: 0.8928571429 + -0.8928571429 = 0.0000000000
0.0000000000 + 2.071428571t + t2 = 0 + -0.8928571429
2.071428571t + t2 = 0 + -0.8928571429

Combine like terms: 0 + -0.8928571429 = -0.8928571429
2.071428571t + t2 = -0.8928571429

The t term is 2.071428571t.  Take half its coefficient (1.035714286).
Square it (1.072704082) and add it to both sides.

Add '1.072704082' to each side of the equation.
2.071428571t + 1.072704082 + t2 = -0.8928571429 + 1.072704082

Reorder the terms:
1.072704082 + 2.071428571t + t2 = -0.8928571429 + 1.072704082

Combine like terms: -0.8928571429 + 1.072704082 = 0.1798469391
1.072704082 + 2.071428571t + t2 = 0.1798469391

Factor a perfect square on the left side:
(t + 1.035714286)(t + 1.035714286) = 0.1798469391

Calculate the square root of the right side: 0.424083646

Break this problem into two subproblems by setting 
(t + 1.035714286) equal to 0.424083646 and -0.424083646.

Subproblem 1

t + 1.035714286 = 0.424083646 Simplifying t + 1.035714286 = 0.424083646 Reorder the terms: 1.035714286 + t = 0.424083646 Solving 1.035714286 + t = 0.424083646 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.035714286' to each side of the equation. 1.035714286 + -1.035714286 + t = 0.424083646 + -1.035714286 Combine like terms: 1.035714286 + -1.035714286 = 0.000000000 0.000000000 + t = 0.424083646 + -1.035714286 t = 0.424083646 + -1.035714286 Combine like terms: 0.424083646 + -1.035714286 = -0.61163064 t = -0.61163064 Simplifying t = -0.61163064

Subproblem 2

t + 1.035714286 = -0.424083646 Simplifying t + 1.035714286 = -0.424083646 Reorder the terms: 1.035714286 + t = -0.424083646 Solving 1.035714286 + t = -0.424083646 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.035714286' to each side of the equation. 1.035714286 + -1.035714286 + t = -0.424083646 + -1.035714286 Combine like terms: 1.035714286 + -1.035714286 = 0.000000000 0.000000000 + t = -0.424083646 + -1.035714286 t = -0.424083646 + -1.035714286 Combine like terms: -0.424083646 + -1.035714286 = -1.459797932 t = -1.459797932 Simplifying t = -1.459797932

Solution

The solution to the problem is based on the solutions from the subproblems. t = {-0.61163064, -1.459797932}

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