Find The \[$ X \$\]-intercept(s) For \[$ F(x) = 6(x-5)^2 - 24 \$\].
Introduction
In mathematics, the x-intercept of a function is the point where the graph of the function crosses the x-axis. This occurs when the value of the function is equal to zero. In this article, we will focus on finding the x-intercept(s) of a quadratic function in the form of f(x) = a(x - h)^2 + k, where a, h, and k are constants.
Understanding the Given Function
The given function is f(x) = 6(x - 5)^2 - 24. This is a quadratic function in the form of f(x) = a(x - h)^2 + k, where a = 6, h = 5, and k = -24.
Finding the x-Intercept(s)
To find the x-intercept(s) of the function, we need to set the function equal to zero and solve for x. This is because the x-intercept occurs when the value of the function is equal to zero.
f(x) = 6(x - 5)^2 - 24 = 0
Step 1: Expand the Squared Term
The first step is to expand the squared term (x - 5)^2. This can be done using the formula (a - b)^2 = a^2 - 2ab + b^2.
(x - 5)^2 = x^2 - 10x + 25
Step 2: Substitute the Expanded Term
Now, substitute the expanded term back into the original equation.
f(x) = 6(x^2 - 10x + 25) - 24 = 0
Step 3: Distribute the Coefficient
The next step is to distribute the coefficient 6 to the terms inside the parentheses.
f(x) = 6x^2 - 60x + 150 - 24 = 0
Step 4: Combine Like Terms
Combine the like terms on the right-hand side of the equation.
f(x) = 6x^2 - 60x + 126 = 0
Step 5: Factor the Quadratic Expression
The final step is to factor the quadratic expression on the left-hand side of the equation. However, in this case, the quadratic expression does not factor easily. Therefore, we will use the quadratic formula to solve for x.
Using the Quadratic Formula
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 6, b = -60, and c = 126.
x = (60 ± √((-60)^2 - 4(6)(126))) / 2(6)
Simplifying the Expression
Simplify the expression inside the square root.
x = (60 ± √(3600 - 3024)) / 12
x = (60 ± √576) / 12
x = (60 ± 24) / 12
Solving for x
Now, solve for x by considering both the positive and negative cases.
Case 1: x = (60 + 24) / 12
x = 84 / 12
x = 7
Case 2: x = (60 - 24) / 12
x = 36 / 12
x = 3
Conclusion
In this article, we found the x-intercept(s) of the quadratic function f(x) = 6(x - 5)^2 - 24. The x-intercept(s) occur at x = 7 and x = 3. These values represent the points where the graph of the function crosses the x-axis.
Final Answer
Q: What is the x-intercept of a function?
A: The x-intercept of a function is the point where the graph of the function crosses the x-axis. This occurs when the value of the function is equal to zero.
Q: How do I find the x-intercept(s) of a quadratic function?
A: To find the x-intercept(s) of a quadratic function, you need to set the function equal to zero and solve for x. This can be done using algebraic methods, such as factoring or the quadratic formula.
Q: What is the quadratic formula?
A: The quadratic formula is a mathematical formula used to solve quadratic equations of the form ax^2 + bx + c = 0. The formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
Q: How do I use the quadratic formula to find the x-intercept(s) of a quadratic function?
A: To use the quadratic formula, you need to identify the values of a, b, and c in the quadratic equation. Then, plug these values into the quadratic formula and simplify the expression to find the x-intercept(s).
Q: What are the steps to find the x-intercept(s) of a quadratic function?
A: The steps to find the x-intercept(s) of a quadratic function are:
- Set the function equal to zero.
- Expand any squared terms.
- Distribute any coefficients.
- Combine like terms.
- Factor the quadratic expression (if possible).
- Use the quadratic formula to solve for x.
Q: Can I use factoring to find the x-intercept(s) of a quadratic function?
A: Yes, you can use factoring to find the x-intercept(s) of a quadratic function. However, factoring may not always be possible, especially for quadratic expressions that do not factor easily.
Q: What are some common mistakes to avoid when finding the x-intercept(s) of a quadratic function?
A: Some common mistakes to avoid when finding the x-intercept(s) of a quadratic function include:
- Not setting the function equal to zero.
- Not expanding squared terms.
- Not distributing coefficients.
- Not combining like terms.
- Not using the correct values of a, b, and c in the quadratic formula.
Q: How do I check my work when finding the x-intercept(s) of a quadratic function?
A: To check your work, plug the x-intercept(s) back into the original function to ensure that the value of the function is equal to zero.
Q: What are some real-world applications of finding the x-intercept(s) of a quadratic function?
A: Some real-world applications of finding the x-intercept(s) of a quadratic function include:
- Modeling the trajectory of a projectile.
- Finding the maximum or minimum value of a quadratic function.
- Determining the x-intercept(s) of a quadratic function in a scientific or engineering context.
Conclusion
In this article, we have answered some frequently asked questions about finding the x-intercept(s) of a quadratic function. We have covered topics such as the x-intercept, quadratic formula, and steps to find the x-intercept(s). We hope that this article has been helpful in clarifying any confusion and providing a better understanding of this important mathematical concept.