Understanding Zeros in a Quadratic Function

Oct 13, 2025 by miodeya.com 113 views

Introduction

In algebra, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. These functions are commonly represented in the form of f(x) = ax^2 + bx + c, where a, b, and c are constants. One of the key characteristics of a quadratic function is its ability to have zeros, which are the values of x that make the function equal to zero. In this article, we will explore how to determine which function has zeros at specific values of x.

Understanding Zeros in a Quadratic Function

A zero of a quadratic function is a value of x that makes the function equal to zero. In other words, if f(x) = 0, then x is a zero of the function. To find the zeros of a quadratic function, we can use the fact that a quadratic function can be factored into the product of two binomials. The zeros of the function are the values of x that make each binomial equal to zero.

Factoring a Quadratic Function

To factor a quadratic function, we need to find two binomials whose product is equal to the original function. The general form of a quadratic function is f(x) = ax^2 + bx + c. To factor this function, we need to find two numbers whose product is ac and whose sum is b. These numbers are the roots of the quadratic equation ax^2 + bx + c = 0.

Using the Factor Theorem

The factor theorem states that if f(x) is a polynomial and f(a) = 0, then (x - a) is a factor of f(x). This means that if we know that a is a zero of the function, then we can write the function as f(x) = (x - a)g(x), where g(x) is another polynomial.

Applying the Factor Theorem to the Given Functions

Now, let's apply the factor theorem to the given functions to determine which one has zeros at x = -2 and x = 5.

Function A: f(x) = x^2 + 2x - 10

To determine if this function has zeros at x = -2 and x = 5, we can plug these values into the function and see if the result is zero.

f(-2) = (-2)^2 + 2(-2) - 10 f(-2) = 4 - 4 - 10 f(-2) = -10 ≠ 0

f(5) = (5)^2 + 2(5) - 10 f(5) = 25 + 10 - 10 f(5) = 25 ≠ 0

Since neither of these values makes the function equal to zero, we can conclude that function A does not have zeros at x = -2 and x = 5.

Function B: f(x) = x^2 - 2x - 10

To determine if this function has zeros at x = -2 and x = 5, we can plug these values into the function and see if the result is zero.

f(-2) = (-2)^2 - 2(-2) - 10 f(-2) = 4 + 4 - 10 f(-2) = -2 ≠ 0

f(5) = (5)^2 - 2(5) - 10 f(5) = 25 - 10 - 10 f(5) = 5 ≠ 0

Since neither of these values makes the function equal to zero, we can conclude that function B does not have zeros at x = -2 and x = 5.

Function C: f(x) = x^2 + 3x - 10

To determine if this function has zeros at x = -2 and x = 5, we can plug these values into the function and see if the result is zero.

f(-2) = (-2)^2 + 3(-2) - 10 f(-2) = 4 - 6 - 10 f(-2) = -12 ≠ 0

f(5) = (5)^2 + 3(5) - 10 f(5) = 25 + 15 - 10 f(5) = 30 ≠ 0

Since neither of these values makes the function equal to zero, we can conclude that function C does not have zeros at x = -2 and x = 5.

Function D: f(x) = x^2 - 3x - 10

To determine if this function has zeros at x = -2 and x = 5, we can plug these values into the function and see if the result is zero.

f(-2) = (-2)^2 - 3(-2) - 10 f(-2) = 4 + 6 - 10 f(-2) = 0

f(5) = (5)^2 - 3(5) - 10 f(5) = 25 - 15 - 10 f(5) = 0

Since both of these values make the function equal to zero, we can conclude that function D has zeros at x = -2 and x = 5.

Conclusion

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. These functions are commonly represented in the form of f(x) = ax^2 + bx + c, where a, b, and c are constants.

Q: What is the difference between a zero and a root of a quadratic function?

A: A zero of a quadratic function is a value of x that makes the function equal to zero. A root of a quadratic function is a value of x that makes the function equal to zero, but it can also be a complex number.

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

A: To find the zeros of a quadratic function, you can use the fact that a quadratic function can be factored into the product of two binomials. The zeros of the function are the values of x that make each binomial equal to zero.

Q: What is the factor theorem?

A: The factor theorem states that if f(x) is a polynomial and f(a) = 0, then (x - a) is a factor of f(x). This means that if we know that a is a zero of the function, then we can write the function as f(x) = (x - a)g(x), where g(x) is another polynomial.

Q: How do I apply the factor theorem to a quadratic function?

A: To apply the factor theorem to a quadratic function, you need to plug in the value of x that you think is a zero of the function into the function and see if the result is zero. If it is, then you can write the function as f(x) = (x - a)g(x), where g(x) is another polynomial.

Q: What is the difference between a quadratic function and a linear function?

A: A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one. This means that a quadratic function has a highest power of two, while a linear function has a highest power of one.

Q: Can a quadratic function have more than two zeros?

A: No, a quadratic function can only have two zeros. This is because a quadratic function can be factored into the product of two binomials, and each binomial can only have one zero.

Q: Can a quadratic function have complex zeros?

A: Yes, a quadratic function can have complex zeros. This is because a quadratic function can be factored into the product of two binomials, and each binomial can have complex zeros.

Q: How do I find the zeros of a quadratic function with complex coefficients?

A: To find the zeros of a quadratic function with complex coefficients, you can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. This formula will give you the two complex zeros of the function.

Q: What is the significance of the zeros of a quadratic function?

A: The zeros of a quadratic function are significant because they represent the values of x that make the function equal to zero. This means that the zeros of a quadratic function are the points where the graph of the function intersects the x-axis.

Q: Can the zeros of a quadratic function be used to solve a system of equations?

A: Yes, the zeros of a quadratic function can be used to solve a system of equations. This is because the zeros of a quadratic function represent the values of x that make the function equal to zero, and these values can be used to solve a system of equations.

Q: How do I use the zeros of a quadratic function to solve a system of equations?

A: To use the zeros of a quadratic function to solve a system of equations, you need to plug in the values of x that you think are the zeros of the function into the system of equations and see if the result is a solution to the system. If it is, then you can use the zeros of the quadratic function to solve the system of equations.