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

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 the quadratic function $f(x) = -2x^2 - 3x + 20$.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means the highest power of the variable $x$ is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In our case, the function is $f(x) = -2x^2 - 3x + 20$, where $a = -2$, $b = -3$, and $c = 20$.

Finding the $x$-Intercepts

To find the $x$-intercepts of the function, we need to set the function equal to zero and solve for $x$. In other words, we need to find the values of $x$ for which $f(x) = 0$. This can be done by factoring the quadratic expression or by using the quadratic formula.

Factoring the Quadratic Expression

The quadratic expression $-2x^2 - 3x + 20$ can be factored as follows:

$$-2x^2 - 3x + 20 = (-2x + 5)(x - 4)$$

Setting the factored expression equal to zero, we get:

$$(-2x + 5)(x - 4) = 0$$

This equation can be solved by setting each factor equal to zero and solving for $x$. This gives us two possible solutions:

$$-2x + 5 = 0 \Rightarrow x = \frac{5}{2}$$

$$x - 4 = 0 \Rightarrow x = 4$$

Therefore, the $x$-intercepts of the function are $\left(\frac{5}{2}, 0\right)$ and $(4, 0)$.

Using the Quadratic Formula

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

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

In our case, $a = -2$, $b = -3$, and $c = 20$. Plugging these values into the quadratic formula, we get:

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

Simplifying the expression, we get:

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

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

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

This gives us two possible solutions:

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

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

Therefore, the $x$-intercepts of the function are $\left(-4, 0\right)$ and $\left(\frac{5}{2}, 0\right)$.

Conclusion

In this article, we have explored how to find the $x$-intercepts of the quadratic function $f(x) = -2x^2 - 3x + 20$. We have used both factoring and the quadratic formula to find the solutions to the equation. The $x$-intercepts of the function are $\left(-4, 0\right)$ and $\left(\frac{5}{2}, 0\right)$.

Final Answer

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

In our previous article, we explored how to find the $x$-intercepts of the quadratic function $f(x) = -2x^2 - 3x + 20$. In this article, we will answer some common questions related to finding the $x$-intercepts of a quadratic function.

Q: What is the $x$-intercept of a quadratic function?

A: The $x$-intercept of a quadratic function is the point where the graph of the function crosses the $x$-axis. In other words, it is the value of $x$ for which the function evaluates to zero.

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

A: There are two common methods to find the $x$-intercepts of a quadratic function: factoring and using the quadratic formula. Factoring involves expressing the quadratic expression as a product of two binomials, while the quadratic formula involves using a mathematical formula to find the solutions to the equation.

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 quadratic formula is given by:

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

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

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 $x$. The solutions to the equation will give you the $x$-intercepts of the function.

Q: What are some common mistakes to avoid when finding the $x$-intercepts of a quadratic function?

A: Some common mistakes to avoid when finding the $x$-intercepts of a quadratic function include:

• Not factoring the quadratic expression correctly
• Not using the correct values of $a$, $b$, and $c$ in the quadratic formula
• Not simplifying the expression correctly
• Not solving for $x$ correctly

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

A: Yes, you can use a calculator to find the $x$-intercepts of a quadratic function. Many calculators have a built-in quadratic formula function that you can use to find the solutions to the equation.

Q: How do I check my work when finding the $x$-intercepts of a quadratic function?

A: To check your work, you can plug the solutions back into the original equation to make sure they are true. You can also graph the function and check if the $x$-intercepts match the solutions you found.

Q: What are some real-world applications of finding the $x$-intercepts of a quadratic function?

A: Finding the $x$-intercepts of a quadratic function has many real-world applications, including:

• Modeling the trajectory of a projectile
• Finding the maximum or minimum value of a quadratic function
• Determining the stability of a system
• Optimizing a quadratic function

Conclusion

In this article, we have answered some common questions related to finding the $x$-intercepts of a quadratic function. We have covered topics such as factoring, the quadratic formula, and common mistakes to avoid. We have also discussed real-world applications of finding the $x$-intercepts of a quadratic function.

Final Answer

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