Solve by Completing the Square 2y^2+14y=-19

Problem

2*y2+14*y=−19

Solution

1. Divide the entire equation by the leading coefficient 2 to make the y2 coefficient equal to 1

y2+7*y=−19/2

2. Identify the coefficient of the linear term y which is 7

3. Calculate the value needed to complete the square by taking half of the linear coefficient and squaring it: (7/2)2=49/4

4. Add 49/4 to both sides of the equation to maintain equality.

y2+7*y+49/4=−19/2+49/4

5. Simplify the right side by finding a common denominator.

−19/2+49/4=−38/4+49/4=11/4

6. Factor the left side as a perfect square trinomial.

(y+7/2)2=11/4

7. Take the square root of both sides, remembering to include both the positive and negative roots.

y+7/2=±√(,11/4)

8. Simplify the radical on the right side.

y+7/2=±√(,11)/2

9. Isolate y by subtracting 7/2 from both sides.

y=−7/2±√(,11)/2

Final Answer

y=(−7±√(,11))/2

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