What Are The \[$x\$\]-intercepts Of The Function \[$f(x) = -2x^2 - 3x + 20\$\]?A. \[$(-4,0)\$\] And \[$\left(\frac{5}{2}, 0\right)\$\]B. \[$\left(-\frac{5}{2}, 0\right)\$\] And \[$(4,0)\$\]C.

What are the $x$-intercepts of the function $f(x) = -2x^2 - 3x + 20$?

Understanding the Concept of $x$-Intercepts

The $x$-intercepts of a function are the points where the graph of the function crosses the $x$-axis. In other words, they are the values of $x$ for which the function evaluates to zero. In this article, we will explore how to find the $x$-intercepts of a quadratic function, specifically the function $f(x) = -2x^2 - 3x + 20$.

The Quadratic Formula

To find the $x$-intercepts of a quadratic function, we can use the quadratic formula. The quadratic formula is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

where $a$, $b$, and $c$ are the coefficients of the quadratic function. In this case, we have:

$$f(x) = -2x^2 - 3x + 20$$

Comparing this with the general form of a quadratic function, $ax^2 + bx + c$, we can see that $a = -2$, $b = -3$, and $c = 20$.

Applying the Quadratic Formula

Now that we have identified the values of $a$, $b$, and $c$, we can plug them into the quadratic formula to find the $x$-intercepts.

$$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(-2)(20)}}{2(-2)}$$

Simplifying the expression, we get:

$$x = \frac{3 \pm \sqrt{9 + 160}}{-4}$$

$$x = \frac{3 \pm \sqrt{169}}{-4}$$

$$x = \frac{3 \pm 13}{-4}$$

Therefore, we have two possible values for $x$:

$$x = \frac{3 + 13}{-4} = -\frac{16}{4} = -4$$

$$x = \frac{3 - 13}{-4} = \frac{-10}{-4} = \frac{5}{2}$$

Conclusion

In conclusion, the $x$-intercepts of the function $f(x) = -2x^2 - 3x + 20$ are $(-4, 0)$ and $\left(\frac{5}{2}, 0\right)$. These are the points where the graph of the function crosses the $x$-axis.

Answer

The correct answer is A. $(-4, 0)$ and $\left(\frac{5}{2}, 0\right)$.

Discussion

The quadratic formula is a powerful tool for finding the $x$-intercepts of a quadratic function. However, it can be challenging to apply, especially when dealing with complex numbers. In this case, we were able to find the $x$-intercepts using the quadratic formula, but we may have encountered difficulties if the expression under the square root had been negative.

Real-World Applications

The concept of $x$-intercepts has many real-world applications. For example, in physics, the $x$-intercepts of a quadratic function can represent the position of an object at a given time. In economics, the $x$-intercepts of a quadratic function can represent the equilibrium price and quantity of a good.

Conclusion

In conclusion, the $x$-intercepts of a quadratic function can be found using the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations, and it has many real-world applications. In this article, we explored how to find the $x$-intercepts of the function $f(x) = -2x^2 - 3x + 20$ using the quadratic formula.

Final Answer

The final answer is A. $(-4, 0)$ and $\left(\frac{5}{2}, 0\right)$.
Q&A: Finding $x$-Intercepts of Quadratic Functions

Understanding the Concept of $x$-Intercepts

The $x$-intercepts of a function are the points where the graph of the function crosses the $x$-axis. In other words, they are the values of $x$ for which the function evaluates to zero. In this article, we will explore how to find the $x$-intercepts of a quadratic function, specifically the function $f(x) = -2x^2 - 3x + 20$.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to find the solutions to a quadratic equation. It is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

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

Q: How do I apply the quadratic formula to find the $x$-intercepts of a quadratic function?

A: To apply the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic function. Then, you can plug these values into the quadratic formula to find the $x$-intercepts.

Q: What are the $x$-intercepts of the function $f(x) = -2x^2 - 3x + 20$?

A: The $x$-intercepts of the function $f(x) = -2x^2 - 3x + 20$ are $(-4, 0)$ and $\left(\frac{5}{2}, 0\right)$. These are the points where the graph of the function crosses the $x$-axis.

Q: How do I know if the quadratic formula will give me real or complex solutions?

A: The quadratic formula will give you real solutions if the expression under the square root is positive. If the expression under the square root is negative, the quadratic formula will give you complex solutions.

Q: What are some real-world applications of the concept of $x$-intercepts?

A: The concept of $x$-intercepts has many real-world applications. For example, in physics, the $x$-intercepts of a quadratic function can represent the position of an object at a given time. In economics, the $x$-intercepts of a quadratic function can represent the equilibrium price and quantity of a good.

Q: Can I use the quadratic formula to find the $x$-intercepts of any quadratic function?

A: Yes, you can use the quadratic formula to find the $x$-intercepts of any quadratic function. However, you need to make sure that the coefficients of the quadratic function are real numbers.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

• Not identifying the values of $a$, $b$, and $c$ correctly
• Not plugging the values of $a$, $b$, and $c$ into the quadratic formula correctly
• Not simplifying the expression under the square root correctly
• Not checking if the expression under the square root is positive or negative

Conclusion

In conclusion, the quadratic formula is a powerful tool for finding the $x$-intercepts of a quadratic function. By understanding how to apply the quadratic formula and avoiding common mistakes, you can use this formula to solve a wide range of quadratic equations.

Final Answer

The final answer is A. $(-4, 0)$ and $\left(\frac{5}{2}, 0\right)$.