Solving Zeros Of Quadratic Function F(x) = 9x² + 6x + 1

Introduction to Finding Zeros of Quadratic Functions

In mathematics, finding the zeros of a quadratic function is a fundamental task with wide-ranging applications. Quadratic functions, characterized by the general form f(x) = ax² + bx + c, where a, b, and c are constants, represent parabolas when graphed. The zeros of a quadratic function are the x-values where the parabola intersects the x-axis, i.e., where f(x) = 0. These zeros are also known as roots or solutions of the quadratic equation ax² + bx + c = 0. Understanding how to find these zeros is crucial for solving various problems in algebra, calculus, and other areas of mathematics and science. For instance, in physics, quadratic functions can model projectile motion, and finding the zeros helps determine the time the projectile hits the ground. In engineering, quadratic equations are used in designing structures and optimizing processes. This article will provide a detailed, step-by-step guide on how to solve for the zeros of a specific quadratic function, $f(x) = 9x^2 + 6x + 1$, ensuring a clear understanding of the process and the underlying principles. We will use the quadratic formula, a powerful tool for finding the zeros of any quadratic equation, and express the answer as a fraction, as requested. By the end of this discussion, you will have a solid grasp of how to apply the quadratic formula and interpret the results in the context of quadratic functions.

Step-by-Step Solution Using the Quadratic Formula

The quadratic formula is a cornerstone in solving quadratic equations, offering a direct method to find the zeros of any quadratic function in the form f(x) = ax² + bx + c. This formula is derived from the process of completing the square and is universally applicable, regardless of the nature of the roots (real, complex, distinct, or repeated). The quadratic formula is expressed as:

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

In this formula, a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. The term inside the square root, b² - 4ac, is known as the discriminant, which plays a critical role in determining the nature of the roots. If the discriminant is positive, the quadratic equation has two distinct real roots; if it is zero, the equation has one real repeated root; and if it is negative, the equation has two complex conjugate roots. Now, let’s apply the quadratic formula to the given function, $f(x) = 9x^2 + 6x + 1$. In this case, we have a = 9, b = 6, and c = 1. Substituting these values into the quadratic formula, we get:

$x = \frac{-6 \pm \sqrt{(6)^2 - 4(9)(1)}}{2(9)}$

This step is crucial as it sets up the equation for simplification and solution. It involves careful substitution of the coefficients into the correct places in the formula, ensuring accuracy in the subsequent steps. The next steps will involve simplifying the expression under the square root and then further simplifying the entire expression to find the values of x that are the zeros of the function. By following this methodical approach, we can confidently find the solutions to any quadratic equation using the quadratic formula.

Applying the Formula to the Given Function

To find the zeros of the quadratic function $f(x) = 9x^2 + 6x + 1$, we begin by identifying the coefficients a, b, and c in the standard quadratic form ax² + bx + c. In this case, a = 9, b = 6, and c = 1. The quadratic formula, which is given by:

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

provides a direct method for finding the roots of any quadratic equation. Substituting the values of a, b, and c into the formula, we have:

$x = \frac{-6 \pm \sqrt{(6)^2 - 4(9)(1)}}{2(9)}$

This equation represents the initial setup for solving the quadratic function. The next step involves simplifying the expression under the square root, which is the discriminant, and then further simplifying the entire expression to find the values of x. The discriminant, b² - 4ac, plays a crucial role in determining the nature of the roots. If it is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, there are two complex roots. By carefully substituting the coefficients into the quadratic formula, we ensure a solid foundation for the subsequent calculations, leading us to the accurate solutions for the zeros of the function. This methodical approach is essential for mastering the application of the quadratic formula in solving various quadratic equations.

Simplifying the Expression

Once we have substituted the coefficients into the quadratic formula, the next crucial step is to simplify the expression. We start by simplifying the term under the square root, which is the discriminant. From the previous step, we have:

$x = \frac{-6 \pm \sqrt{(6)^2 - 4(9)(1)}}{2(9)}$

Now, let’s simplify the discriminant:

$(6)^2 - 4(9)(1) = 36 - 36 = 0$

So, the expression under the square root simplifies to 0. This means that the quadratic equation has exactly one real root (a repeated root). Now, we substitute this value back into the quadratic formula:

$x = \frac{-6 \pm \sqrt{0}}{18}$

Since the square root of 0 is 0, the equation further simplifies to:

$x = \frac{-6 \pm 0}{18}$

This means that both the plus and minus cases will yield the same result, indicating a single, repeated root. The next step involves performing the final simplification to find the value of x. By carefully simplifying each part of the expression, we reduce the complexity and move closer to finding the precise solution for the zeros of the quadratic function. This methodical simplification process is a key skill in solving quadratic equations and ensuring accuracy in the final answer.

Calculating the Zeros

After simplifying the expression obtained from the quadratic formula, we are now in a position to calculate the zeros of the quadratic function. From the previous steps, we have:

$x = \frac{-6 \pm 0}{18}$

Since adding or subtracting 0 does not change the value, we can simplify this to:

$x = \frac{-6}{18}$

Now, we reduce the fraction to its simplest form. Both the numerator and the denominator are divisible by 6. Dividing both by 6, we get:

$x = \frac{-6 \div 6}{18 \div 6} = \frac{-1}{3}$

Thus, the quadratic function $f(x) = 9x^2 + 6x + 1$ has one real root, which is $x = -\frac{1}{3}$. This means that the parabola represented by the quadratic function touches the x-axis at only one point, $x = -\frac{1}{3}$. In the context of the quadratic formula, a discriminant of 0 indicates that the quadratic equation has a single, repeated root, which we have now found to be $-\frac{1}{3}$. This calculation demonstrates the power of the quadratic formula in finding the zeros of quadratic functions and highlights the importance of simplifying expressions to arrive at the final answer. By following these steps, we have successfully solved for the zeros of the given quadratic function and expressed the answer as a fraction.

Expressing the Answer as a Fraction

The final step in solving for the zeros of the quadratic function $f(x) = 9x^2 + 6x + 1$ is to express the answer as a fraction. As we have seen in the previous steps, after applying the quadratic formula and simplifying the expression, we arrived at:

$x = \frac{-6}{18}$

To express this as a fraction in its simplest form, we need to reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 6 and 18 is 6. Dividing both the numerator and the denominator by 6, we get:

$x = \frac{-6 \div 6}{18 \div 6} = \frac{-1}{3}$

Therefore, the zero of the quadratic function $f(x) = 9x^2 + 6x + 1$ is $x = -\frac{1}{3}$. This is already expressed as a fraction, and it is in its simplest form. The negative sign indicates that the root is on the negative side of the x-axis. Expressing the answer as a fraction is crucial for precision, especially in mathematical contexts where decimal approximations may not be accurate enough. In this case, the fraction $-\frac{1}{3}$ provides the exact value of the zero, which is essential for further calculations or analysis involving this quadratic function. By ensuring that the final answer is expressed as a simplified fraction, we maintain mathematical rigor and clarity in our solution.

Conclusion: Understanding Zeros of Quadratic Functions

In conclusion, solving for the zeros of a quadratic function is a fundamental skill in algebra with numerous applications in mathematics and various scientific fields. Through this detailed, step-by-step solution for the quadratic function $f(x) = 9x^2 + 6x + 1$, we have demonstrated the effective use of the quadratic formula and the importance of simplifying expressions to arrive at the correct answer. The quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, is a powerful tool that allows us to find the zeros of any quadratic equation in the form ax² + bx + c = 0. By identifying the coefficients a, b, and c and substituting them into the formula, we can systematically solve for the values of x that make the function equal to zero. The discriminant, b² - 4ac, plays a crucial role in determining the nature of the roots, indicating whether there are two distinct real roots, one real repeated root, or two complex roots. In the case of $f(x) = 9x^2 + 6x + 1$, the discriminant was zero, indicating one real repeated root. We simplified the expression obtained from the quadratic formula and found the zero to be $x = -\frac{1}{3}$, expressing it as a fraction in its simplest form. This process highlights the importance of precision and mathematical rigor in finding solutions. Understanding the zeros of quadratic functions is essential for graphing parabolas, solving optimization problems, and modeling real-world phenomena. This comprehensive guide provides a solid foundation for tackling more complex quadratic equations and related mathematical concepts.