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Chapter 9: Problem 82

Consider the function $$f(x)=x^{2}-4 x+3$$ Find the zeros of the function.

Short Answer

Expert verified
The zeros of the function are \( x = 1 \) and \( x = 3 \).

Step by step solution

01

- Identify the function

The given function is \[ f(x) = x^2 - 4x + 3 \] We need to find the zeros of this function, which are the values of \(x\) where \( f(x) = 0 \).
02

- Set the function equal to zero

To find the zeros, set the function equal to zero: \[ x^2 - 4x + 3 = 0 \]
03

- Factor the quadratic equation

Factor the quadratic equation on the left-hand side. We are looking for two numbers that multiply to 3 and add up to -4. These numbers are -1 and -3. So, the equation factors as: \[ (x - 1)(x - 3) = 0 \]
04

- Solve for x

Set each factor equal to zero: \[ x - 1 = 0 \] \[ x - 3 = 0 \] Solving each equation for \( x \) gives: \[ x = 1 \] \[ x = 3 \]
05

- Write the solution

The zeros of the function \( f(x) = x^2 - 4x + 3 \) are \( x = 1 \) and \( x = 3 \).

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Key Concepts

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

Zeros of Functions
To understand the zeros of a function, think of them as the points where the graph of the function touches or crosses the x-axis. In simpler terms, the zeros are the values of x that make the function equal to zero.
For a given function f(x), the zeros are found by setting the function equal to zero and solving for x. For instance, for the function \( f(x) = x^2 - 4x + 3 \), we find the zeros by solving the equation \(x^2 - 4x + 3 = 0 \).
Identifying the zeros of a function helps in understanding the function鈥檚 behavior and its graph.
Factoring Quadratics
Factoring quadratics is a method used to break down quadratic expressions into their simpler components. A quadratic expression is usually in the form of \(ax^2 + bx + c \).
In the case of \( f(x) = x^2 - 4x + 3 \), we look for two numbers that multiply to the constant term (3) and add up to the coefficient of the linear term (-4).
Those numbers are -1 and -3. So, we can express the quadratic as: \((x - 1)(x - 3) = 0 \)
Factoring makes it simpler to solve quadratic equations by turning them into a product of factors set to zero, which can then be solved individually.
Solving Equations
Once you have factored the quadratic equation, the next step is to solve for the variable. For instance, from our factored equation: \( (x - 1)(x - 3) = 0 \), we set each factor to zero and solve separately.
This provides us with two equations:
- \( x - 1 = 0 \)
- \( x - 3 = 0 \)
Solving these, we get:
  • \( x = 1 \)
  • \( x = 3 \)
These values are the solutions or roots of the quadratic equation. It's important to check our work by substituting these values back into the original equation to ensure they make the equation true.

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Most popular questions from this chapter

Nancy jogs and walks to campus each day. She averages \(4 \mathrm{km} / \mathrm{h}\) walking and \(8 \mathrm{km} / \mathrm{h}\) jogging. The distance from home to the campus is \(6 \mathrm{km},\) and she makes the trip in 1 hr. How far does she jog on each trip?

Suppose that it takes 12 units of carbohydrates and 6 units of protein to satisfy Jacob's minimum weekly requirements. A particular type of meat contains 2 units of carbohydrates and 2 units of protein per pound. A particular cheese contains 3 units of carbohydrates and 1 unit of protein per pound. The meat costs 3.50 Ddollar per pound and the cheese costs \(\$ 4.60\) per pound. How many pounds of each are needed in order to minimize the cost and still meet the minimum requirements?

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Find two different row-echelon forms of $$ \left[\begin{array}{ll} 1 & 5 \\ 3 & 2 \end{array}\right] $$
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