To find which of the given options is equivalent to the function \( f(x) = x^2 - 6x + 8 \), we can perform some algebraic manipulations. We can start by factoring or completing the square.
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Factoring: We look for two numbers that multiply to \( 8 \) (the constant term) and add to \(-6\) (the coefficient of \( x \)). The numbers \(-2\) and \(-4\) work because: \[ -2 \cdot -4 = 8 \quad \text{and} \quad -2 + (-4) = -6 \] Thus, we can factor the quadratic: \[ f(x) = (x - 2)(x - 4) \]
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Completing the square: We can rewrite the function in vertex form. Start with the original function: \[ f(x) = x^2 - 6x + 8 \] We take half of \(-6\), square it, and rearrange: \[ f(x) = (x^2 - 6x + 9) - 9 + 8 \] This simplifies to: \[ f(x) = (x - 3)^2 - 1 \]
From the options given, we can see the following:
- \( f(x) = (x - 6)^2 - 28 \) is not equivalent.
- \( f(x) = (x - 2)(x - 4) \) is equivalent (as shown in our factoring step).
- \( f(x) = (x - 3)(x - 3) \) simplifies to \( (x - 3)^2 \), which does not equal the original function.
- \( f(x) = (x - 3)^2 - 9 \) is also not equivalent since it doesn't produce the original quadratic.
Thus, the correct answer is:
f(x) = (x−2)(x−4).