Question
Solve the equation: \( (x+3)^{2}+5(x+3)-6=0 \)
Ask by West Cervantes. in the United States
Dec 11,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solutions are \( x=-9 \) and \( x=-2 \).
Solution
Solve the equation \( (x+3)^{2}+5(x+3)-6=0 \).
Solve the quadratic equation by following steps:
- step0: Solve by factoring:
\(\left(x+3\right)^{2}+5\left(x+3\right)-6=0\)
- step1: Expand the expression:
\(x^{2}+11x+18=0\)
- step2: Factor the expression:
\(\left(x+2\right)\left(x+9\right)=0\)
- step3: Separate into possible cases:
\(\begin{align}&x+2=0\\&x+9=0\end{align}\)
- step4: Solve the equation:
\(\begin{align}&x=-2\\&x=-9\end{align}\)
- step5: Rewrite:
\(x_{1}=-9,x_{2}=-2\)
The solutions to the equation \( (x+3)^{2}+5(x+3)-6=0 \) are \( x=-9 \) and \( x=-2 \).
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Simplify this solution The Deep Dive
Let's simplify the equation by substituting \(y = x + 3\). Then the equation becomes: \( y^2 + 5y - 6 = 0 \) Now, we can factor this quadratic equation: \((y + 6)(y - 1) = 0\) This gives us two solutions: 1. \( y + 6 = 0 \) → \( y = -6 \) 2. \( y - 1 = 0 \) → \( y = 1 \) Now, revert back to \(x\): 1. For \(y = -6\): \(x + 3 = -6\) → \(x = -9\) 2. For \(y = 1\): \(x + 3 = 1\) → \(x = -2\) Thus, the solutions are \(x = -9\) and \(x = -2\).
