Question
Write the equation of a quadratic function with the given solutions. Use standard form \( f(x)=a x^{2}+b x+c \). \( x=2 \) and \( x=1 \)
Ask by Paul Cervantes. in the United States
Mar 15,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The quadratic function with roots \( x=2 \) and \( x=1 \) in standard form is:
\[
f(x) = x^2 - 3x + 2
\]
Solution
1. The quadratic function with roots \( x=2 \) and \( x=1 \) can be written in factored form as:
\[
f(x)=a(x-2)(x-1)
\]
where \( a \) is any non-zero constant.
2. To convert this into standard form \( f(x)=ax^2+bx+c \), we first expand the factors:
\[
f(x)=a\left[(x-2)(x-1)\right]=a\left[x^2 - x - 2x + 2\right]=a\left[x^2 - 3x + 2\right]
\]
3. Distributing the constant \( a \) gives:
\[
f(x)=ax^2 - 3ax + 2a
\]
4. Choosing \( a=1 \) for simplicity, the quadratic function becomes:
\[
f(x)=x^2-3x+2
\]
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Simplify this solution The Deep Dive
To create a quadratic function with the given solutions \( x=2 \) and \( x=1 \), you can start with the factored form: \( f(x) = a(x - 2)(x - 1) \). Expanding this gives: \[ f(x) = a(x^2 - 3x + 2) \] Now, let’s express this in standard form: \[ f(x) = a x^2 - 3a x + 2a \] Setting \( a = 1 \) for simplicity, we get: \[ f(x) = x^2 - 3x + 2 \] This is the quadratic function in standard form. So, in essence, you can easily plug in any value of \( a \) to stretch or compress your parabola, while keeping the roots intact!
