Question
Solve for \( y \). \[ y^{2}-y-2=0 \] If there is more than one solution, separate them with commas. If there is no solution, click on "No solution." \( y=\square \)
Ask by Kirk Harper. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( y = 2, -1 \)
Solution
1. Start with the quadratic equation:
\[
y^2 - y - 2 = 0
\]
2. Factor the quadratic by finding two numbers that multiply to \(-2\) and add to \(-1\). The numbers \(-2\) and \(1\) satisfy these conditions:
\[
y^2 - y - 2 = (y - 2)(y + 1)
\]
3. Set each factor equal to zero and solve for \( y \):
\[
y - 2 = 0 \quad \text{or} \quad y + 1 = 0
\]
\[
y = 2 \quad \text{or} \quad y = -1
\]
4. The solutions are:
\[
y = 2, -1
\]
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Simplify this solution Extra Insights
To solve the equation \( y^{2}-y-2=0 \), we can factor it. We need two numbers that multiply to \(-2\) and add to \(-1\). These numbers are \(-2\) and \(1\). Thus, we can factor the equation as: \[ (y-2)(y+1)=0 \] Setting each factor to zero gives us the solutions: \[ y-2=0 \quad \Rightarrow \quad y=2 \] \[ y+1=0 \quad \Rightarrow \quad y=-1 \] So, the solutions are: \( y=2, -1 \)
