Finding The Zeros Of Quadratic Function F(x) = 4x^2 + 24x + 11

by Sharif Sakr 63 views

Hey guys! Let's dive into the fascinating world of quadratic functions and explore how to find their zeros. Today, we're tackling the quadratic function $f(x) = 4x^2 + 24x + 11$. Our mission is to determine which of the given options—A. $x = -9.25$, B. $x = -5.5$, C. $x = 0.5$, or D. $x = 3.25$—is a zero of this function. So, buckle up and let's get started!

Understanding Quadratic Functions and Zeros

Before we jump into solving the problem, let's make sure we're all on the same page about what quadratic functions and their zeros are. A quadratic function is a polynomial function of the second degree, generally written in the form $f(x) = ax^2 + bx + c$, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, which is a U-shaped curve. The zeros of a quadratic function, also known as the roots or x-intercepts, are the values of x for which the function equals zero, i.e., $f(x) = 0$. These are the points where the parabola intersects the x-axis. Finding the zeros is a fundamental concept in algebra and has numerous applications in various fields, including physics, engineering, and economics.

In the context of our problem, finding the zeros of the quadratic function $f(x) = 4x^2 + 24x + 11$ means we need to solve the equation $4x^2 + 24x + 11 = 0$. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. We'll explore these methods and determine which one is most suitable for our specific problem.

Methods to Find the Zeros of a Quadratic Function

There are several methods to find the zeros of a quadratic function, each with its own strengths and weaknesses. Let's discuss the most common methods:

1. Factoring

Factoring involves expressing the quadratic equation as a product of two binomials. If we can factor the quadratic equation $ax^2 + bx + c = 0$ into the form $(px + q)(rx + s) = 0$, then the zeros are the solutions to the equations $px + q = 0$ and $rx + s = 0$. Factoring is often the quickest method, but it's not always easy to find the factors, especially when the coefficients are large or the roots are not rational numbers. To apply factoring, we need to find two numbers that multiply to $a imes c$ and add up to b. In our case, $a = 4$, $b = 24$, and $c = 11$, so we need to find two numbers that multiply to $4 imes 11 = 44$ and add up to 24. While this method can be efficient, it might not be immediately obvious how to factor this particular equation. Therefore, let's consider alternative methods.

2. Completing the Square

Completing the square is a method that transforms the quadratic equation into a perfect square trinomial, which can then be easily solved. This method is particularly useful when the quadratic equation is not easily factorable. To complete the square for the equation $ax^2 + bx + c = 0$, we first divide the equation by a (if a is not 1), then move the constant term to the right side of the equation. Next, we add the square of half the coefficient of the x term to both sides of the equation. This creates a perfect square trinomial on the left side, which can be factored as $(x + k)^2$, where k is half the coefficient of the x term. Finally, we take the square root of both sides and solve for x. While completing the square is a reliable method, it can be a bit more time-consuming than factoring or using the quadratic formula, especially when dealing with fractions.

3. Quadratic Formula

The quadratic formula is a general formula that provides the solutions to any quadratic equation of the form $ax^2 + bx + c = 0$. The formula is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation. The quadratic formula is a powerful tool because it works for all quadratic equations, regardless of whether they are factorable or not. It's especially useful when the roots are irrational or complex numbers. To use the quadratic formula, we simply plug in the values of $a$, $b$, and $c$ from our equation and simplify. This method is generally the most straightforward and reliable, especially when dealing with equations that are difficult to factor or complete the square. Given the structure of our equation and the need for a precise answer, the quadratic formula seems like the most efficient approach.

Applying the Quadratic Formula to $f(x) = 4x^2 + 24x + 11$

Now, let's apply the quadratic formula to our function $f(x) = 4x^2 + 24x + 11$. Here, we have $a = 4$, $b = 24$, and $c = 11$. Plugging these values into the quadratic formula, we get:

x = rac{-24 ext{±} ext{√}(24^2 - 4 imes 4 imes 11)}{2 imes 4}

Let's simplify this step by step:

  1. Calculate $24^2$: $24^2 = 576$
  2. Calculate $4 imes 4 imes 11$: $4 imes 4 imes 11 = 176$
  3. Subtract the result from step 2 from the result in step 1: $576 - 176 = 400$
  4. Find the square root of the result from step 3: $ ext{√}400 = 20$
  5. Multiply $2$ by $4$ in the denominator: $2 imes 4 = 8$

Now, our equation looks like this:

x = rac{-24 ext{±} 20}{8}

We have two possible solutions, one with addition and one with subtraction:

Solution 1: Using Addition

x_1 = rac{-24 + 20}{8} = rac{-4}{8} = -0.5

Solution 2: Using Subtraction

x_2 = rac{-24 - 20}{8} = rac{-44}{8} = -5.5

So, the zeros of the quadratic function $f(x) = 4x^2 + 24x + 11$ are $x = -0.5$ and $x = -5.5$.

Comparing Our Solutions with the Given Options

Now that we've found the zeros, let's compare them with the options provided:

A. $x = -9.25$ B. $x = -5.5$ C. $x = 0.5$ D. $x = 3.25$

We found that the zeros are $x = -0.5$ and $x = -5.5$. Option B, $x = -5.5$, matches one of our solutions. Therefore, the correct answer is B.

Conclusion

In conclusion, by applying the quadratic formula to the function $f(x) = 4x^2 + 24x + 11$, we found the zeros to be $x = -0.5$ and $x = -5.5$. Comparing these results with the given options, we determined that option B, $x = -5.5$, is a zero of the function. This exercise highlights the power and versatility of the quadratic formula in solving quadratic equations. Remember, understanding the different methods for finding zeros, like factoring, completing the square, and using the quadratic formula, can help you tackle a wide range of problems in algebra and beyond. Keep practicing, and you'll become a pro at solving quadratic equations in no time! And that's how we nail it, guys!