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Chapter 6: Problem 51

Find the zeros of each function. $$ f(x)=x^{2}-4 x-45 $$

Short Answer

Expert verified
The zeros of the function are x = 9 and x = -5.

Step by step solution

01

Understand the problem

To find the zeros of the function, we need to determine the values of x for which the function equals zero: \( f(x) = x^2 - 4x - 45 = 0 \)
02

Factor the quadratic equation

Look for two numbers that multiply to the constant term (-45) and add up to the coefficient of the linear term (-4). Those numbers are 9 and -5: \( x^2 - 4x - 45 = (x - 9)(x + 5) \)
03

Set each factor to zero

Set each factor equal to zero and solve for x: \( x - 9 = 0 \) Therefore, \( x = 9 \). \( x + 5 = 0 \) Therefore, \( x = -5 \).
04

Verify the solutions

Substitute the values of x back into the original function to ensure they indeed make the function zero. For \( x = 9 \): \( f(9) = 9^2 - 4(9) - 45 = 81 - 36 - 45 = 0 \) For \( x = -5 \): \( f(-5) = (-5)^2 - 4(-5) - 45 = 25 + 20 - 45 = 0 \)

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Key Concepts

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

Quadratic Equations
A quadratic equation is a polynomial equation of degree 2. It typically has the form x = ax^2 + bx + c = 0 where 'a', 'b', and 'c' are constants. Quadratic equations can have one, two, or no real solutions. Solving quadratic equations is a fundamental skill in algebra. To solve them:
  • you can use factoring if the equation can be factored
  • you can use the quadratic formula: x = [-b 卤 sqrt(b^2 - 4ac)] / 2a
  • you can complete the square
Quadratic equations are essential in various fields, including physics, engineering, and economics. They often model real-world scenarios such as projectile motion and area optimization.
Factoring
Factoring is a method used to solve quadratic equations by expressing the quadratic expression as a product of two binomials. In our example, we start with the quadratic function f(x) = x^2 - 4x - 45 . The steps for factoring include:
  • Identifying two numbers that multiply to the constant term (-45) and add to the linear coefficient (-4). These numbers are 9 and -5.
  • Writing the factored form: (x - 9)(x + 5)
Factoring is like reversing multiplication and is handy when solving quadratic equations and finding zeros of functions. Practicing factoring is crucial since it simplifies many algebraic problems.
Solving Equations
Solving equations involves finding the values of variables that make the equation true. For quadratic equations, once we have the factored form, we can easily find solutions. Using our example:
  • Set each factor equal to zero: x - 9 = 0 and x + 5 = 0
  • Solve for x: x = 9 and x = -5
Finally, verifying solutions is important to ensure the accuracy. Substitute the values back into the original equation: f(9) = 81 - 36 - 45 = 0 and f(-5) = 25 + 20 - 45 = 0
This confirms that both x = 9 and x = -5 are correct solutions.

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Most popular questions from this chapter

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 9 x^{4}-25 x^{2} $$

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