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Math  /  Algebra

QuestionComplete the square and find the vertex form of the quadratic function. f(x)=x2−8x+59f(x)=□\begin{array}{l} f(x)=x^{2}-8 x+59 \\ f(x)=\square \end{array}

Studdy Solution

STEP 1

1. The given quadratic function is f(x)=x2−8x+59 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 f(x) = a(x-h)^2 + k , where (h,k)(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 xx, which is −8-8. To complete the square, take half of this coefficient and square it:
(−82)2=(−4)2=16 \left(\frac{-8}{2}\right)^2 = (-4)^2 = 16
Add and subtract this square inside the function to maintain equality:
f(x)=x2−8x+16−16+59 f(x) = x^2 - 8x + 16 - 16 + 59

STEP 4

Rewrite the expression by grouping the perfect square trinomial and simplifying the constants:
f(x)=(x2−8x+16)+(−16+59) f(x) = (x^2 - 8x + 16) + (-16 + 59)
The perfect square trinomial x2−8x+16x^2 - 8x + 16 can be written as:
(x−4)2 (x - 4)^2
Now simplify the constants:
−16+59=43 -16 + 59 = 43
Thus, the function becomes:
f(x)=(x−4)2+43 f(x) = (x - 4)^2 + 43
This is the vertex form of the quadratic function, where the vertex is (4,43)(4, 43).

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