Question
Question Use the square roots property to solve the quadratic equation \( w^{2}+6 w+9=121 \). If there are multiple answers, list them separated by a comma, e.g. 1,2 . If there is no solution, enter \( \varnothing \).
Ask by Wright Willis. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solutions are 8, -14.
Solution
Rewrite the quadratic as (w+3)² since w² + 6w + 9 factors as (w+3)². The equation becomes:
(w+3)² = 121
Take the square root of both sides, remembering to include both the positive and negative roots:
w + 3 = ±11
Solve for w in each case:
1. If w + 3 = 11, then w = 11 - 3 = 8.
2. If w + 3 = -11, then w = -11 - 3 = -14.
Thus, the solutions are 8, -14.
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Simplify this solution The Deep Dive
First, let's rewrite the equation by moving 121 to the left side: \( w^{2} + 6w + 9 - 121 = 0 \) This simplifies to: \( w^{2} + 6w - 112 = 0 \) Next, we can recognize that the left side can be factored: \( (w + 12)(w - 6) = 0 \) From here, we can set each factor equal to zero: 1. \( w + 12 = 0 \) gives \( w = -12 \) 2. \( w - 6 = 0 \) gives \( w = 6 \) Thus, the solutions to the equation are: -12, 6
