Math Snap

STEP 1

1. The given quadratic function is $f(x) = x^2 - 8x + 59$.
2. We need to rewrite this function in vertex form by completing the square.
3. The vertex form of a quadratic function is $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex.

STEP 2

1. Identify the coefficient of the linear term and use it to complete the square.
2. Rewrite the quadratic expression in vertex form.

STEP 3

Identify the coefficient of the linear term $x$, which is $-8$. To complete the square, take half of this coefficient and square it:
$$\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16$$ Add and subtract this square inside the function to maintain equality:
$$f(x) = x^2 - 8x + 16 - 16 + 59$$

Rewrite the expression by grouping the perfect square trinomial and simplifying the constants:
$$f(x) = (x^2 - 8x + 16) + (-16 + 59)$$ The perfect square trinomial $x^2 - 8x + 16$ can be written as:
$$(x - 4)^2$$ Now simplify the constants:
$$-16 + 59 = 43$$ Thus, the function becomes:
$$f(x) = (x - 4)^2 + 43$$ This is the vertex form of the quadratic function, where the vertex is $(4, 43)$.