10y^2-15y-7=0

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Solution for 10y^2-15y-7=0 equation:


Simplifying
10y2 + -15y + -7 = 0

Reorder the terms:
-7 + -15y + 10y2 = 0

Solving
-7 + -15y + 10y2 = 0

Solving for variable 'y'.

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

Divide each side by '10'.
-0.7 + -1.5y + y2 = 0

Move the constant term to the right:

Add '0.7' to each side of the equation.
-0.7 + -1.5y + 0.7 + y2 = 0 + 0.7

Reorder the terms:
-0.7 + 0.7 + -1.5y + y2 = 0 + 0.7

Combine like terms: -0.7 + 0.7 = 0.0
0.0 + -1.5y + y2 = 0 + 0.7
-1.5y + y2 = 0 + 0.7

Combine like terms: 0 + 0.7 = 0.7
-1.5y + y2 = 0.7

The y term is -1.5y.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5y + 0.5625 + y2 = 0.7 + 0.5625

Reorder the terms:
0.5625 + -1.5y + y2 = 0.7 + 0.5625

Combine like terms: 0.7 + 0.5625 = 1.2625
0.5625 + -1.5y + y2 = 1.2625

Factor a perfect square on the left side:
(y + -0.75)(y + -0.75) = 1.2625

Calculate the square root of the right side: 1.123610253

Break this problem into two subproblems by setting 
(y + -0.75) equal to 1.123610253 and -1.123610253.

Subproblem 1

y + -0.75 = 1.123610253 Simplifying y + -0.75 = 1.123610253 Reorder the terms: -0.75 + y = 1.123610253 Solving -0.75 + y = 1.123610253 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = 1.123610253 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = 1.123610253 + 0.75 y = 1.123610253 + 0.75 Combine like terms: 1.123610253 + 0.75 = 1.873610253 y = 1.873610253 Simplifying y = 1.873610253

Subproblem 2

y + -0.75 = -1.123610253 Simplifying y + -0.75 = -1.123610253 Reorder the terms: -0.75 + y = -1.123610253 Solving -0.75 + y = -1.123610253 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = -1.123610253 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = -1.123610253 + 0.75 y = -1.123610253 + 0.75 Combine like terms: -1.123610253 + 0.75 = -0.373610253 y = -0.373610253 Simplifying y = -0.373610253

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.873610253, -0.373610253}

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