What Are The Zeros Of The Equation $y = (x^2 - 81)(x + 4)$?A. X = − 9 , − 4 , 9 X = -9, -4, 9 X = − 9 , − 4 , 9 B. X = − 9 , − 4 X = -9, -4 X = − 9 , − 4 OnlyC. X = − 7 , 9 X = -7, 9 X = − 7 , 9 D. X = − 9 , 4 X = -9, 4 X = − 9 , 4 Only

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Introduction

In mathematics, a zero of a function is a value of the variable that makes the function equal to zero. In other words, it is a solution to the equation when the function is set equal to zero. In this article, we will focus on finding the zeros of the equation $y = (x^2 - 81)(x + 4)$. This equation represents a quadratic function, and we will use various mathematical techniques to solve for the zeros.

Understanding the Equation

The given equation is a quadratic function in the form of $y = (x^2 - 81)(x + 4)$. To find the zeros, we need to set the function equal to zero and solve for the variable x. The equation can be rewritten as $y = x^3 + 4x^2 - 81x - 324$. However, this form is not necessary to find the zeros, as we can use the factored form of the equation to our advantage.

Factoring the Equation

The equation is already factored as $y = (x^2 - 81)(x + 4)$. This means that we can set each factor equal to zero and solve for x. The first factor, $x^2 - 81$, can be set equal to zero as follows:

x2−81=0x^2 - 81 = 0

Solving the First Factor

To solve the equation $x^2 - 81 = 0$, we can add 81 to both sides of the equation, resulting in:

x2=81x^2 = 81

Taking the square root of both sides of the equation, we get:

x=±81x = \pm \sqrt{81}

x=±9x = \pm 9

Solving the Second Factor

The second factor, $x + 4$, can be set equal to zero as follows:

x+4=0x + 4 = 0

Subtracting 4 from both sides of the equation, we get:

x=−4x = -4

Finding the Zeros

Now that we have solved both factors, we can find the zeros of the equation by combining the solutions. The zeros of the equation are the values of x that make the function equal to zero. In this case, the zeros are:

x=−9,−4,9x = -9, -4, 9

Conclusion

In conclusion, the zeros of the equation $y = (x^2 - 81)(x + 4)$ are $x = -9, -4, 9$. These values of x make the function equal to zero, and they represent the solutions to the equation.

Answer Key

The correct answer is:

A. x=−9,−4,9x = -9, -4, 9

Discussion

The equation $y = (x^2 - 81)(x + 4)$ is a quadratic function that can be factored into two linear factors. By setting each factor equal to zero and solving for x, we can find the zeros of the equation. The zeros of the equation are the values of x that make the function equal to zero, and they represent the solutions to the equation.

Tips and Tricks

  • When solving a quadratic equation, it is often helpful to factor the equation into two linear factors.
  • By setting each factor equal to zero and solving for x, we can find the zeros of the equation.
  • The zeros of the equation are the values of x that make the function equal to zero, and they represent the solutions to the equation.

Related Topics

  • Quadratic equations
  • Factoring quadratic equations
  • Zeros of a function
  • Solutions to an equation

Further Reading

  • For more information on quadratic equations, see the article "Quadratic Equations: A Comprehensive Guide".
  • For more information on factoring quadratic equations, see the article "Factoring Quadratic Equations: A Step-by-Step Guide".
  • For more information on zeros of a function, see the article "Zeros of a Function: A Comprehensive Guide".

Introduction

In our previous article, we discussed how to find the zeros of a quadratic equation. In this article, we will provide a Q&A guide to help you understand the concept of quadratic equation zeros and how to solve them.

Q: What are the zeros of a quadratic equation?

A: The zeros of a quadratic equation are the values of the variable that make the equation equal to zero. In other words, they are the solutions to the equation when the function is set equal to zero.

Q: How do I find the zeros of a quadratic equation?

A: To find the zeros of a quadratic equation, you can use various methods such as factoring, the quadratic formula, or graphing. Factoring is often the easiest method, especially for simple quadratic equations.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that can be used to find the solutions to a quadratic equation. The formula is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where a, b, and c are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula to find the zeros of a quadratic equation?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. Then, simplify the expression and solve for x.

Q: What are the advantages and disadvantages of using the quadratic formula?

A: The advantages of using the quadratic formula are that it can be used to find the solutions to any quadratic equation, regardless of whether it can be factored or not. However, the formula can be complex and difficult to work with, especially for large values of a, b, and c.

Q: Can I use the quadratic formula to find the zeros of a quadratic equation with complex solutions?

A: Yes, the quadratic formula can be used to find the zeros of a quadratic equation with complex solutions. However, the solutions will be in the form of complex numbers, which can be represented as a + bi, where a and b are real numbers and i is the imaginary unit.

Q: How do I determine whether a quadratic equation has real or complex solutions?

A: To determine whether a quadratic equation has real or complex solutions, you can use the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

Q: What are some common mistakes to avoid when finding the zeros of a quadratic equation?

A: Some common mistakes to avoid when finding the zeros of a quadratic equation include:

  • Not factoring the equation correctly
  • Not using the correct values of a, b, and c in the quadratic formula
  • Not simplifying the expression correctly
  • Not checking for complex solutions

Q: How can I practice finding the zeros of a quadratic equation?

A: You can practice finding the zeros of a quadratic equation by working through examples and exercises. You can also use online resources, such as calculators and graphing tools, to help you visualize the solutions.

Conclusion

In conclusion, finding the zeros of a quadratic equation is an important concept in algebra. By understanding the different methods for finding the zeros, such as factoring and the quadratic formula, you can solve quadratic equations with ease. Remember to avoid common mistakes and practice regularly to become proficient in finding the zeros of a quadratic equation.

Related Topics

  • Quadratic equations
  • Factoring quadratic equations
  • Quadratic formula
  • Complex solutions
  • Discriminant

Further Reading

  • For more information on quadratic equations, see the article "Quadratic Equations: A Comprehensive Guide".
  • For more information on factoring quadratic equations, see the article "Factoring Quadratic Equations: A Step-by-Step Guide".
  • For more information on the quadratic formula, see the article "The Quadratic Formula: A Guide to Solving Quadratic Equations".