If { X = 3 $}$ Is A Zero Of The Function Below, What Is The Value Of { B $} ? ? ? { F(x) = X^2 + B X + 24 \} ${ B = }$
Introduction
In algebra, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. In this article, we will explore how to find the value of b in a quadratic function given that x = 3 is a zero of the function.
What are Zeros of a Function?
A zero of a 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. Zeros are also known as roots or solutions of the function.
Given Information
We are given that x = 3 is a zero of the function f(x) = x^2 + bx + 24. This means that when x = 3, the function f(x) is equal to zero.
Substituting x = 3 into the Function
To find the value of b, we can substitute x = 3 into the function f(x) = x^2 + bx + 24.
f(3) = (3)^2 + b(3) + 24 f(3) = 9 + 3b + 24
Simplifying the Equation
Since x = 3 is a zero of the function, we know that f(3) = 0. Therefore, we can set up the equation:
9 + 3b + 24 = 0
Combining Like Terms
We can combine the constant terms on the left-hand side of the equation:
33 + 3b = 0
Isolating the Variable
To isolate the variable b, we can subtract 33 from both sides of the equation:
3b = -33
Solving for b
Finally, we can divide both sides of the equation by 3 to solve for b:
b = -33/3 b = -11
Conclusion
In this article, we have shown how to find the value of b in a quadratic function given that x = 3 is a zero of the function. We substituted x = 3 into the function, simplified the equation, and isolated the variable b to find the value of b. The final answer is b = -11.
Example Problems
- If x = 2 is a zero of the function f(x) = x^2 + bx + 15, what is the value of b?
- If x = 5 is a zero of the function f(x) = x^2 + bx + 20, what is the value of b?
Step-by-Step Solutions
- To find the value of b, we can substitute x = 2 into the function f(x) = x^2 + bx + 15.
f(2) = (2)^2 + b(2) + 15 f(2) = 4 + 2b + 15
Since x = 2 is a zero of the function, we know that f(2) = 0. Therefore, we can set up the equation:
4 + 2b + 15 = 0
Combining like terms, we get:
19 + 2b = 0
Isolating the variable b, we get:
2b = -19
Solving for b, we get:
b = -19/2 b = -9.5
- To find the value of b, we can substitute x = 5 into the function f(x) = x^2 + bx + 20.
f(5) = (5)^2 + b(5) + 20 f(5) = 25 + 5b + 20
Since x = 5 is a zero of the function, we know that f(5) = 0. Therefore, we can set up the equation:
25 + 5b + 20 = 0
Combining like terms, we get:
45 + 5b = 0
Isolating the variable b, we get:
5b = -45
Solving for b, we get:
b = -45/5 b = -9
Final Answer
Introduction
In our previous article, we explored how to find the value of b in a quadratic function given that x = 3 is a zero of the function. In this article, we will answer some frequently asked questions about quadratic function zeros.
Q: What is a zero of a function?
A: A zero of a 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.
Q: How do I find the zeros of a quadratic function?
A: To find the zeros of a quadratic function, you can use the factoring method, the quadratic formula, or the graphing method. The factoring method involves factoring the quadratic expression into the product of two binomials, while the quadratic formula involves using the formula x = (-b ± √(b^2 - 4ac)) / 2a to find the zeros.
Q: What is the quadratic formula?
A: The quadratic formula is a formula that is used to find the zeros of a quadratic function. The formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic expression.
Q: How do I use the quadratic formula to find the zeros of a quadratic function?
A: To use the quadratic formula to find the zeros of a quadratic function, you need to identify the coefficients a, b, and c of the quadratic expression. Then, you can plug these values into the quadratic formula to find the zeros.
Q: What is the difference between a zero and a root of a function?
A: A zero and a root of a function are the same thing. They refer to a value of x that makes the function equal to zero.
Q: Can a quadratic function have more than two zeros?
A: No, a quadratic function can have at most two zeros. This is because a quadratic function is a polynomial of degree two, and a polynomial of degree two can have at most two zeros.
Q: How do I find the value of b in a quadratic function given that x = 3 is a zero of the function?
A: To find the value of b in a quadratic function given that x = 3 is a zero of the function, you can substitute x = 3 into the function and solve for b. This involves setting up an equation using the function and the fact that x = 3 is a zero of the function.
Q: What is the value of b in the quadratic function f(x) = x^2 + bx + 24 given that x = 3 is a zero of the function?
A: To find the value of b in the quadratic function f(x) = x^2 + bx + 24 given that x = 3 is a zero of the function, you can substitute x = 3 into the function and solve for b. This involves setting up an equation using the function and the fact that x = 3 is a zero of the function.
f(3) = (3)^2 + b(3) + 24 f(3) = 9 + 3b + 24
Since x = 3 is a zero of the function, we know that f(3) = 0. Therefore, we can set up the equation:
9 + 3b + 24 = 0
Combining like terms, we get:
33 + 3b = 0
Isolating the variable b, we get:
3b = -33
Solving for b, we get:
b = -33/3 b = -11
Conclusion
In this article, we have answered some frequently asked questions about quadratic function zeros. We have discussed what a zero of a function is, how to find the zeros of a quadratic function, and how to use the quadratic formula to find the zeros of a quadratic function. We have also provided an example of how to find the value of b in a quadratic function given that x = 3 is a zero of the function.
Example Problems
- If x = 2 is a zero of the function f(x) = x^2 + bx + 15, what is the value of b?
- If x = 5 is a zero of the function f(x) = x^2 + bx + 20, what is the value of b?
Step-by-Step Solutions
- To find the value of b, we can substitute x = 2 into the function f(x) = x^2 + bx + 15.
f(2) = (2)^2 + b(2) + 15 f(2) = 4 + 2b + 15
Since x = 2 is a zero of the function, we know that f(2) = 0. Therefore, we can set up the equation:
4 + 2b + 15 = 0
Combining like terms, we get:
19 + 2b = 0
Isolating the variable b, we get:
2b = -19
Solving for b, we get:
b = -19/2 b = -9.5
- To find the value of b, we can substitute x = 5 into the function f(x) = x^2 + bx + 20.
f(5) = (5)^2 + b(5) + 20 f(5) = 25 + 5b + 20
Since x = 5 is a zero of the function, we know that f(5) = 0. Therefore, we can set up the equation:
25 + 5b + 20 = 0
Combining like terms, we get:
45 + 5b = 0
Isolating the variable b, we get:
5b = -45
Solving for b, we get:
b = -45/5 b = -9
Final Answer
The final answer is b = -11.