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Simplifying 12y2 + -34y + 16 = 0 Reorder the terms: 16 + -34y + 12y2 = 0 Solving 16 + -34y + 12y2 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '2'. 2(8 + -17y + 6y2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(8 + -17y + 6y2)' equal to zero and attempt to solve: Simplifying 8 + -17y + 6y2 = 0 Solving 8 + -17y + 6y2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 1.333333333 + -2.833333333y + y2 = 0 Move the constant term to the right: Add '-1.333333333' to each side of the equation. 1.333333333 + -2.833333333y + -1.333333333 + y2 = 0 + -1.333333333 Reorder the terms: 1.333333333 + -1.333333333 + -2.833333333y + y2 = 0 + -1.333333333 Combine like terms: 1.333333333 + -1.333333333 = 0.000000000 0.000000000 + -2.833333333y + y2 = 0 + -1.333333333 -2.833333333y + y2 = 0 + -1.333333333 Combine like terms: 0 + -1.333333333 = -1.333333333 -2.833333333y + y2 = -1.333333333 The y term is -2.833333333y. Take half its coefficient (-1.416666667). Square it (2.006944445) and add it to both sides. Add '2.006944445' to each side of the equation. -2.833333333y + 2.006944445 + y2 = -1.333333333 + 2.006944445 Reorder the terms: 2.006944445 + -2.833333333y + y2 = -1.333333333 + 2.006944445 Combine like terms: -1.333333333 + 2.006944445 = 0.673611112 2.006944445 + -2.833333333y + y2 = 0.673611112 Factor a perfect square on the left side: (y + -1.416666667)(y + -1.416666667) = 0.673611112 Calculate the square root of the right side: 0.820738151 Break this problem into two subproblems by setting (y + -1.416666667) equal to 0.820738151 and -0.820738151.Subproblem 1
y + -1.416666667 = 0.820738151 Simplifying y + -1.416666667 = 0.820738151 Reorder the terms: -1.416666667 + y = 0.820738151 Solving -1.416666667 + y = 0.820738151 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.416666667' to each side of the equation. -1.416666667 + 1.416666667 + y = 0.820738151 + 1.416666667 Combine like terms: -1.416666667 + 1.416666667 = 0.000000000 0.000000000 + y = 0.820738151 + 1.416666667 y = 0.820738151 + 1.416666667 Combine like terms: 0.820738151 + 1.416666667 = 2.237404818 y = 2.237404818 Simplifying y = 2.237404818Subproblem 2
y + -1.416666667 = -0.820738151 Simplifying y + -1.416666667 = -0.820738151 Reorder the terms: -1.416666667 + y = -0.820738151 Solving -1.416666667 + y = -0.820738151 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.416666667' to each side of the equation. -1.416666667 + 1.416666667 + y = -0.820738151 + 1.416666667 Combine like terms: -1.416666667 + 1.416666667 = 0.000000000 0.000000000 + y = -0.820738151 + 1.416666667 y = -0.820738151 + 1.416666667 Combine like terms: -0.820738151 + 1.416666667 = 0.595928516 y = 0.595928516 Simplifying y = 0.595928516Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.237404818, 0.595928516}Solution
y = {2.237404818, 0.595928516}
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