Question
\( \left(b^{2}+8\right)^{2}-\left(b^{2}+8\right)-72=0 \) The solution set is \( \{\tilde{\square}\} \).
Ask by Bright Gordon. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solutions are \( b = -1 \) and \( b = 1 \).
Solution
Solve the equation \( (b^{2}+8)^{2}-(b^{2}+8)-72=0 \).
Solve the equation by following steps:
- step0: Solve for \(b\):
\(\left(b^{2}+8\right)^{2}-\left(b^{2}+8\right)-72=0\)
- step1: Subtract the terms:
\(\left(b^{2}+8\right)^{2}-b^{2}-80=0\)
- step2: Calculate:
\(b^{4}+15b^{2}-16=0\)
- step3: Factor the expression:
\(\left(b-1\right)\left(b+1\right)\left(b^{2}+16\right)=0\)
- step4: Separate into possible cases:
\(\begin{align}&b-1=0\\&b+1=0\\&b^{2}+16=0\end{align}\)
- step5: Solve the equation:
\(\begin{align}&b=1\\&b=-1\\&b \notin \mathbb{R}\end{align}\)
- step6: Find the union:
\(\begin{align}&b=1\\&b=-1\end{align}\)
- step7: Rewrite:
\(b_{1}=-1,b_{2}=1\)
The solution set for the equation \( (b^{2}+8)^{2}-(b^{2}+8)-72=0 \) is \( \{b_{1}=-1,b_{2}=1\} \).
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Simplify this solution Extra Insights
To solve the equation \( \left(b^{2}+8\right)^{2}-\left(b^{2}+8\right)-72=0 \), let’s make a substitution. Let \( x = b^{2} + 8 \). The equation then transforms into \( x^2 - x - 72 = 0 \). We can factor this quadratic equation to find \( (x - 9)(x + 8) = 0 \). This gives us \( x = 9 \) and \( x = -8 \). Since \( x = b^{2} + 8 \), we can only take the solution where \( b^{2} + 8 = 9 \), leading to \( b^{2} = 1 \) and thus \( b = \pm 1 \). The solution set is \( \{-1, 1\} \).
