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Chapter 8: Problem 10

Complete the square to write each function in the form \(f(x)=a(x-h)^{2}+k\). $$f(x)=x^{2}-6 x-1$$

Short Answer

Expert verified
f(x) = (x - 3)^2 - 10

Step by step solution

01

- Identify Coefficients

Identify the coefficients from the quadratic function. Here, the function is given by \( f(x) = x^2 - 6x - 1 \). The coefficients are: \( a = 1 \), \( b = -6 \), and \( c = -1 \).
02

- Complete the Square

To complete the square, add and subtract the square of half the coefficient of \( x \). The coefficient of \( x \) is -6, half of this is -3, and squaring this gives \( (-3)^2 = 9 \). Thus, \( f(x) = x^2 - 6x + 9 - 9 - 1 \).
03

- Rewrite the Expression

Rewrite the quadratic expression by grouping the perfect square and the constants: \( f(x) = (x - 3)^2 - 9 - 1 \).
04

- Simplify

Simplify the constants to obtain the final form of the function: \( f(x) = (x - 3)^2 - 10 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic functions
Quadratic functions are mathematical functions of the form \[ f(x) = ax^2 + bx + c \]. They are called 'quadratic' because they involve the variable squared, or raised to the power of 2.
These functions create a curve on a graph known as a parabola. In this form:
  • \(a\) is the coefficient in front of the squared term and determines how

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