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Simplifying 6y2 + -5y + 6 + -12y2 + -5y + 3 = 0 Reorder the terms: 6 + 3 + -5y + -5y + 6y2 + -12y2 = 0 Combine like terms: 6 + 3 = 9 9 + -5y + -5y + 6y2 + -12y2 = 0 Combine like terms: -5y + -5y = -10y 9 + -10y + 6y2 + -12y2 = 0 Combine like terms: 6y2 + -12y2 = -6y2 9 + -10y + -6y2 = 0 Solving 9 + -10y + -6y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -1.5 + 1.666666667y + y2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 1.666666667y + 1.5 + y2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 1.666666667y + y2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 1.666666667y + y2 = 0 + 1.5 1.666666667y + y2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 1.666666667y + y2 = 1.5 The y term is 1.666666667y. Take half its coefficient (0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. 1.666666667y + 0.6944444447 + y2 = 1.5 + 0.6944444447 Reorder the terms: 0.6944444447 + 1.666666667y + y2 = 1.5 + 0.6944444447 Combine like terms: 1.5 + 0.6944444447 = 2.1944444447 0.6944444447 + 1.666666667y + y2 = 2.1944444447 Factor a perfect square on the left side: (y + 0.8333333335)(y + 0.8333333335) = 2.1944444447 Calculate the square root of the right side: 1.481365736 Break this problem into two subproblems by setting (y + 0.8333333335) equal to 1.481365736 and -1.481365736.Subproblem 1
y + 0.8333333335 = 1.481365736 Simplifying y + 0.8333333335 = 1.481365736 Reorder the terms: 0.8333333335 + y = 1.481365736 Solving 0.8333333335 + y = 1.481365736 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + y = 1.481365736 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + y = 1.481365736 + -0.8333333335 y = 1.481365736 + -0.8333333335 Combine like terms: 1.481365736 + -0.8333333335 = 0.6480324025 y = 0.6480324025 Simplifying y = 0.6480324025Subproblem 2
y + 0.8333333335 = -1.481365736 Simplifying y + 0.8333333335 = -1.481365736 Reorder the terms: 0.8333333335 + y = -1.481365736 Solving 0.8333333335 + y = -1.481365736 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + y = -1.481365736 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + y = -1.481365736 + -0.8333333335 y = -1.481365736 + -0.8333333335 Combine like terms: -1.481365736 + -0.8333333335 = -2.3146990695 y = -2.3146990695 Simplifying y = -2.3146990695Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.6480324025, -2.3146990695}
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