How To Find The X Intercepts Of F(x) = -x^2 - X + 2

To determine the $x$-intercepts of the function $f(x) = -x^2 - x + 2$, we need to find the values of $x$ for which $f(x) = 0$. In other words, we are solving the quadratic equation $-x^2 - x + 2 = 0$. This article will guide you through the process of finding these intercepts and provide a comprehensive understanding of the underlying concepts.

Understanding $x$-intercepts

Before diving into the solution, it’s crucial to understand what $x$-intercepts are. The x-intercepts of a function are the points where the graph of the function intersects the $x$-axis. At these points, the $y$-coordinate (or the function value, $f(x)$) is zero. Finding the $x$-intercepts is a fundamental concept in algebra and calculus, providing valuable insights into the behavior of functions. For quadratic functions, which have the general form $f(x) = ax^2 + bx + c$, the $x$-intercepts represent the roots or solutions of the quadratic equation $ax^2 + bx + c = 0$.

Methods to Find $x$-intercepts

There are several methods to find the $x$-intercepts of a quadratic function, including factoring, using the quadratic formula, and completing the square. For the function $f(x) = -x^2 - x + 2$, we will primarily focus on factoring and using the quadratic formula. Factoring is often the quickest method if the quadratic expression can be easily factored. The quadratic formula is a more general approach that works for any quadratic equation.

Factoring the Quadratic Expression

Our main objective is to solve the equation $-x^2 - x + 2 = 0$. To make factoring easier, we can multiply the entire equation by $-1$ to get rid of the negative sign in front of the $x^2$ term. This gives us:

$x^2 + x - 2 = 0$

Now, we look for two numbers that multiply to $-2$ and add to $1$. These numbers are $2$ and $-1$. Thus, we can factor the quadratic expression as follows:

$(x + 2)(x - 1) = 0$

Setting each factor equal to zero gives us the solutions for $x$:

$x + 2 = 0$ or $x - 1 = 0$

Solving these equations, we find:

$x = -2$ or $x = 1$

Therefore, the $x$-intercepts are $x = -2$ and $x = 1$. These correspond to the points $(-2, 0)$ and $(1, 0)$ on the graph of the function.

Using the Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations of the form $ax^2 + bx + c = 0$. The formula is:

x = rac{-b ext{±} ext{√(}b^2 - 4ac)}{2a}

For our equation $-x^2 - x + 2 = 0$, we have $a = -1$, $b = -1$, and $c = 2$. Plugging these values into the quadratic formula, we get:

x = rac{-(-1) ext{±} ext{√}((-1)^2 - 4(-1)(2))}{2(-1)}

Simplifying, we have:

x = rac{1 ext{±} ext{√}(1 + 8)}{-2}

x = rac{1 ext{±} ext{√}9}{-2}

x = rac{1 ext{±} 3}{-2}

This gives us two possible values for $x$:

x = rac{1 + 3}{-2} = rac{4}{-2} = -2

x = rac{1 - 3}{-2} = rac{-2}{-2} = 1

Again, we find the $x$-intercepts are $x = -2$ and $x = 1$, corresponding to the points $(-2, 0)$ and $(1, 0)$.

Detailed Steps for Factoring

Let's break down the factoring method into more detailed steps. Factoring involves rewriting the quadratic expression as a product of two binomials. This method relies on identifying two numbers that satisfy specific conditions related to the coefficients of the quadratic expression. Here's a step-by-step guide:

1. Rewrite the Equation: Start with the quadratic equation $-x^2 - x + 2 = 0$. Multiply both sides by $-1$ to obtain a positive leading coefficient: $x^2 + x - 2 = 0$.

2. Identify Coefficients: Identify the coefficients $a$, $b$, and $c$ in the quadratic equation $ax^2 + bx + c = 0$. In this case, $a = 1$, $b = 1$, and $c = -2$.

3. Find Two Numbers: Look for two numbers that multiply to $ac$ (which is $1 imes -2 = -2$) and add up to $b$ (which is $1$). These numbers are $2$ and $-1$ because $2 imes -1 = -2$ and $2 + (-1) = 1$.

4. Rewrite the Middle Term: Rewrite the middle term ($+x$) using the two numbers found in the previous step: $x^2 + 2x - x - 2 = 0$.

5. Factor by Grouping: Group the terms in pairs and factor out the greatest common factor (GCF) from each pair:

   $x(x + 2) - 1(x + 2) = 0$

6. Factor Out the Common Binomial: Notice that $(x + 2)$ is a common factor in both terms. Factor it out:

   $(x + 2)(x - 1) = 0$

7. Set Each Factor to Zero: Set each factor equal to zero and solve for $x$:

   $x + 2 = 0$ or $x - 1 = 0$

8. Solve for $x$: Solve each equation to find the $x$-intercepts:

   $x = -2$ or $x = 1$

Thus, the $x$-intercepts are $(-2, 0)$ and $(1, 0)$.

Step-by-Step Guide to Using the Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation, regardless of whether it can be easily factored. This method involves substituting the coefficients of the quadratic equation into a specific formula to find the roots. Here's a detailed step-by-step guide:

1. Identify Coefficients: Start with the quadratic equation in the form $ax^2 + bx + c = 0$. For the equation $-x^2 - x + 2 = 0$, identify the coefficients $a = -1$, $b = -1$, and $c = 2$.

2. Write the Quadratic Formula: Recall the quadratic formula:

   x = rac{-b ext{±} ext{√(}b^2 - 4ac)}{2a}

3. Substitute the Values: Substitute the values of $a$, $b$, and $c$ into the quadratic formula:

   x = rac{-(-1) ext{±} ext{√}((-1)^2 - 4(-1)(2))}{2(-1)}

4. Simplify the Expression: Simplify the expression step by step:

   x = rac{1 ext{±} ext{√}(1 + 8)}{-2}

   x = rac{1 ext{±} ext{√}9}{-2}

   x = rac{1 ext{±} 3}{-2}

5. Solve for the Two Possible Values of $x$: Separate the ± sign into two separate equations to find the two possible values of $x$:

   x = rac{1 + 3}{-2} and x = rac{1 - 3}{-2}

6. Calculate the Values: Calculate each value:

   x = rac{4}{-2} = -2

   x = rac{-2}{-2} = 1

Thus, the $x$-intercepts are $x = -2$ and $x = 1$, corresponding to the points $(-2, 0)$ and $(1, 0)$.

Graphical Interpretation

The x-intercepts have a significant graphical interpretation. They are the points where the parabola, which is the graph of a quadratic function, intersects the $x$-axis. For the function $f(x) = -x^2 - x + 2$, the parabola opens downwards (since the coefficient of $x^2$ is negative) and intersects the $x$-axis at $x = -2$ and $x = 1$. These points are crucial for sketching the graph of the function, as they provide the locations where the function's value is zero. The vertex of the parabola, another key point, lies midway between the $x$-intercepts. In this case, the $x$-coordinate of the vertex is rac{-2 + 1}{2} = -rac{1}{2}.

Common Mistakes to Avoid

When finding $x$-intercepts, it's essential to avoid common mistakes that can lead to incorrect answers. One frequent error is incorrectly applying the quadratic formula, especially with the signs. Ensure that you substitute the values of $a$, $b$, and $c$ correctly and simplify the expression meticulously. Another mistake is incorrectly factoring the quadratic expression. Always double-check your factors by expanding them to see if they match the original quadratic expression. Additionally, some students may confuse $x$-intercepts with $y$-intercepts. Remember, $x$-intercepts are the points where $y = 0$, while the $y$-intercept is the point where $x = 0$.

Practice Problems

To solidify your understanding, let's work through a few practice problems:

1. Find the $x$-intercepts of the function $f(x) = 2x^2 - 6x + 4$.
2. Determine the $x$-intercepts of $g(x) = x^2 + 4x - 5$.
3. What are the $x$-intercepts of $h(x) = -3x^2 + 12x - 9$?

By solving these problems, you can reinforce the methods discussed and enhance your problem-solving skills.

Conclusion

In summary, finding the x-intercepts of the function $f(x) = -x^2 - x + 2$ involves solving the quadratic equation $-x^2 - x + 2 = 0$. We can use factoring or the quadratic formula to find the solutions, which are $x = -2$ and $x = 1$. These correspond to the points $(-2, 0)$ and $(1, 0)$ on the graph of the function. Understanding how to find $x$-intercepts is a critical skill in algebra, with applications in various mathematical contexts and real-world scenarios. By mastering these methods and avoiding common mistakes, you can confidently solve quadratic equations and interpret the results graphically.