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Simplifying 12y2 + -28y + 15 = 1 Reorder the terms: 15 + -28y + 12y2 = 1 Solving 15 + -28y + 12y2 = 1 Solving for variable 'y'. Reorder the terms: 15 + -1 + -28y + 12y2 = 1 + -1 Combine like terms: 15 + -1 = 14 14 + -28y + 12y2 = 1 + -1 Combine like terms: 1 + -1 = 0 14 + -28y + 12y2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(7 + -14y + 6y2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(7 + -14y + 6y2)' equal to zero and attempt to solve: Simplifying 7 + -14y + 6y2 = 0 Solving 7 + -14y + 6y2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 1.166666667 + -2.333333333y + y2 = 0 Move the constant term to the right: Add '-1.166666667' to each side of the equation. 1.166666667 + -2.333333333y + -1.166666667 + y2 = 0 + -1.166666667 Reorder the terms: 1.166666667 + -1.166666667 + -2.333333333y + y2 = 0 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + -2.333333333y + y2 = 0 + -1.166666667 -2.333333333y + y2 = 0 + -1.166666667 Combine like terms: 0 + -1.166666667 = -1.166666667 -2.333333333y + y2 = -1.166666667 The y term is -2.333333333y. Take half its coefficient (-1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. -2.333333333y + 1.361111112 + y2 = -1.166666667 + 1.361111112 Reorder the terms: 1.361111112 + -2.333333333y + y2 = -1.166666667 + 1.361111112 Combine like terms: -1.166666667 + 1.361111112 = 0.194444445 1.361111112 + -2.333333333y + y2 = 0.194444445 Factor a perfect square on the left side: (y + -1.166666667)(y + -1.166666667) = 0.194444445 Calculate the square root of the right side: 0.440958552 Break this problem into two subproblems by setting (y + -1.166666667) equal to 0.440958552 and -0.440958552.Subproblem 1
y + -1.166666667 = 0.440958552 Simplifying y + -1.166666667 = 0.440958552 Reorder the terms: -1.166666667 + y = 0.440958552 Solving -1.166666667 + y = 0.440958552 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + y = 0.440958552 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + y = 0.440958552 + 1.166666667 y = 0.440958552 + 1.166666667 Combine like terms: 0.440958552 + 1.166666667 = 1.607625219 y = 1.607625219 Simplifying y = 1.607625219Subproblem 2
y + -1.166666667 = -0.440958552 Simplifying y + -1.166666667 = -0.440958552 Reorder the terms: -1.166666667 + y = -0.440958552 Solving -1.166666667 + y = -0.440958552 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + y = -0.440958552 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + y = -0.440958552 + 1.166666667 y = -0.440958552 + 1.166666667 Combine like terms: -0.440958552 + 1.166666667 = 0.725708115 y = 0.725708115 Simplifying y = 0.725708115Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.607625219, 0.725708115}Solution
y = {1.607625219, 0.725708115}
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