7y(4y-14)=23

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Solution for 7y(4y-14)=23 equation:


Simplifying
7y(4y + -14) = 23

Reorder the terms:
7y(-14 + 4y) = 23
(-14 * 7y + 4y * 7y) = 23
(-98y + 28y2) = 23

Solving
-98y + 28y2 = 23

Solving for variable 'y'.

Reorder the terms:
-23 + -98y + 28y2 = 23 + -23

Combine like terms: 23 + -23 = 0
-23 + -98y + 28y2 = 0

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

Divide each side by '28'.
-0.8214285714 + -3.5y + y2 = 0

Move the constant term to the right:

Add '0.8214285714' to each side of the equation.
-0.8214285714 + -3.5y + 0.8214285714 + y2 = 0 + 0.8214285714

Reorder the terms:
-0.8214285714 + 0.8214285714 + -3.5y + y2 = 0 + 0.8214285714

Combine like terms: -0.8214285714 + 0.8214285714 = 0.0000000000
0.0000000000 + -3.5y + y2 = 0 + 0.8214285714
-3.5y + y2 = 0 + 0.8214285714

Combine like terms: 0 + 0.8214285714 = 0.8214285714
-3.5y + y2 = 0.8214285714

The y term is -3.5y.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5y + 3.0625 + y2 = 0.8214285714 + 3.0625

Reorder the terms:
3.0625 + -3.5y + y2 = 0.8214285714 + 3.0625

Combine like terms: 0.8214285714 + 3.0625 = 3.8839285714
3.0625 + -3.5y + y2 = 3.8839285714

Factor a perfect square on the left side:
(y + -1.75)(y + -1.75) = 3.8839285714

Calculate the square root of the right side: 1.970768523

Break this problem into two subproblems by setting 
(y + -1.75) equal to 1.970768523 and -1.970768523.

Subproblem 1

y + -1.75 = 1.970768523 Simplifying y + -1.75 = 1.970768523 Reorder the terms: -1.75 + y = 1.970768523 Solving -1.75 + y = 1.970768523 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + y = 1.970768523 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + y = 1.970768523 + 1.75 y = 1.970768523 + 1.75 Combine like terms: 1.970768523 + 1.75 = 3.720768523 y = 3.720768523 Simplifying y = 3.720768523

Subproblem 2

y + -1.75 = -1.970768523 Simplifying y + -1.75 = -1.970768523 Reorder the terms: -1.75 + y = -1.970768523 Solving -1.75 + y = -1.970768523 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + y = -1.970768523 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + y = -1.970768523 + 1.75 y = -1.970768523 + 1.75 Combine like terms: -1.970768523 + 1.75 = -0.220768523 y = -0.220768523 Simplifying y = -0.220768523

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.720768523, -0.220768523}

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