Question
\#8 \( \begin{array}{c}f(x)=x^{2}-8 x+15 \\ x^{2}-8 x+15=0\end{array} \)
Ask by Moran Rojas. in the United States
Mar 17,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solutions are \(x=3\) and \(x=5\).
Solution
Solve the equation \( x^{2}-8x+15=0 \).
Solve the quadratic equation by following steps:
- step0: Solve by factoring:
\(x^{2}-8x+15=0\)
- step1: Factor the expression:
\(\left(x-5\right)\left(x-3\right)=0\)
- step2: Separate into possible cases:
\(\begin{align}&x-5=0\\&x-3=0\end{align}\)
- step3: Solve the equation:
\(\begin{align}&x=5\\&x=3\end{align}\)
- step4: Rewrite:
\(x_{1}=3,x_{2}=5\)
The solutions to the equation \(x^{2}-8x+15=0\) are \(x=3\) and \(x=5\).
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Simplify this solution Bonus Knowledge
To solve the equation \( x^2 - 8x + 15 = 0 \), we can factor it. The equation factors into \( (x-3)(x-5) = 0 \). Therefore, the solutions are \( x = 3 \) and \( x = 5 \). This means the parabola represented by the function f(x) crosses the x-axis at these points. Now let's quickly discuss the vertex of the parabola given by \( f(x) \). The vertex form can be derived by completing the square, yielding the vertex at \( (4, -1) \). This point represents the minimum value of the function, as the parabola opens upwards. Keep it in mind for graphing or interpreting the function's behavior!
