Finding Zeros: F(x)=9x^2+6x+1 As A Fraction

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Finding the Zeros of a Quadratic Function: f(x) = 9x² + 6x + 1

Hey guys! Today, we're diving into a classic math problem: finding the zeros of a quadratic function. Specifically, we're tackling the function f(x) = 9x² + 6x + 1. Not only will we find those elusive zeros, but we'll also express them as fractions, because who doesn't love a good fraction? So, grab your pencils, and let's get started!

Understanding Quadratic Functions

Before we jump into solving, let's briefly recap what quadratic functions are all about. A quadratic function is a polynomial function of degree two, generally written in the form f(x) = ax² + 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, a U-shaped curve that opens either upwards (if a > 0) or downwards (if a < 0). In our case, f(x) = 9x² + 6x + 1, we have a = 9, b = 6, and c = 1. Since a is positive, our parabola opens upwards.

The zeros of a quadratic function, also known as the roots or x-intercepts, are the values of x for which f(x) = 0. In other words, they are the points where the parabola intersects the x-axis. Finding these zeros is a fundamental problem in algebra, and there are several methods we can use to solve it, including factoring, completing the square, and using the quadratic formula. Each method has its own advantages and disadvantages, depending on the specific quadratic function we're dealing with. For some quadratics, factoring is straightforward, while others require the quadratic formula. Understanding these different approaches allows us to choose the most efficient method for each problem, making our mathematical journey smoother and more enjoyable. In this particular case, we'll explore factoring as it turns out to be the most direct route to our solution. So, let's keep this in mind as we go through the steps.

Method 1: Factoring the Quadratic

In our case, the quadratic function f(x) = 9x² + 6x + 1 is actually a perfect square trinomial. This means it can be factored into the form (px + q)², where p and q are constants. Recognizing this pattern can save us a lot of time and effort. Let's break down how to factor it:

  1. Recognize the Perfect Square Trinomial: A perfect square trinomial has the form a² + 2ab + b² or a² - 2ab + b². In our function, 9x² is (3x)², 1 is 1², and 6x is 2(3x)1. So, we can see that it fits the pattern a² + 2ab + b² with a = 3x and b = 1.
  2. Factor the Trinomial: Since we've identified it as a perfect square trinomial, we can factor it as (3x + 1)². This means f(x) = (3x + 1)(3x + 1).
  3. Set the Factor Equal to Zero: To find the zeros, we set the factored form equal to zero: (3x + 1)² = 0. This simplifies to 3x + 1 = 0.
  4. Solve for x: Now, we solve for x: 3x = -1, which gives us x = -1/3. Since both factors are the same, we only have one unique zero. This makes sense graphically as the parabola just touches the x-axis at one point.

Therefore, the zero of the quadratic function f(x) = 9x² + 6x + 1 is x = -1/3. And there you have it, a fraction as promised!

Method 2: Using the Quadratic Formula

If factoring isn't your cup of tea, or if the quadratic function isn't easily factorable, we can always rely on the quadratic formula. This formula provides a general solution for finding the zeros of any quadratic equation in the form ax² + bx + c = 0. The formula is:

x = (-b ± √(b² - 4ac)) / (2a)

Let's apply this to our function, f(x) = 9x² + 6x + 1:

  1. Identify a, b, and c: As we mentioned earlier, a = 9, b = 6, and c = 1.

  2. Plug the Values into the Formula:

    x = (-6 ± √(6² - 4 * 9 * 1)) / (2 * 9)

    x = (-6 ± √(36 - 36)) / 18

    x = (-6 ± √0) / 18

    x = -6 / 18

  3. Simplify: x = -1/3

As you can see, the quadratic formula gives us the same zero, x = -1/3. The fact that the discriminant (b² - 4ac) is zero indicates that the quadratic has exactly one real root, which confirms our earlier finding.

Method 3: Completing the Square

Completing the square is another powerful technique for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily solved. While it might seem a bit more involved than factoring or using the quadratic formula, it provides a deeper understanding of the structure of quadratic equations. Let's see how it works for our function, f(x) = 9x² + 6x + 1:

  1. Rewrite the Equation: Start with the equation 9x² + 6x + 1 = 0.

  2. Factor out the Leading Coefficient (if necessary): In this case, the coefficient of x² is already a perfect square. In some cases, you may need to factor out the leading coefficient of x² so that the x² term has a coefficient of 1. However, since 9 is a perfect square, we can avoid that step here.

  3. Move the Constant Term to the Other Side: Subtract 1 from both sides: 9x² + 6x = -1.

  4. Complete the Square: To complete the square, we need to add (b / 2a)² to both sides of the equation. In our case, b = 6 and a = 9, so we have to be a little careful. However, because we recognized that our original expression was a perfect square, we can jump to rewriting the left side:

    (3x+1)² = 0

  5. Take the Square Root of Both Sides: Taking the square root of both sides gives us 3x + 1 = 0.

  6. Solve for x: 3x = -1, so x = -1/3.

Again, we arrive at the same zero, x = -1/3. Completing the square reinforces the idea that our quadratic function is a perfect square trinomial, making it clear why we only have one real root.

Verification

To ensure our solution is correct, we can plug x = -1/3 back into the original function:

f(-1/3) = 9(-1/3)² + 6(-1/3) + 1

f(-1/3) = 9(1/9) - 2 + 1

f(-1/3) = 1 - 2 + 1

f(-1/3) = 0

Since f(-1/3) = 0, we have verified that x = -1/3 is indeed a zero of the function f(x) = 9x² + 6x + 1.

Conclusion

So, there you have it! We successfully found the zero of the quadratic function f(x) = 9x² + 6x + 1 and expressed it as a fraction: x = -1/3. We explored multiple methods, including factoring, using the quadratic formula, and completing the square, each leading us to the same result. Factoring turned out to be the most straightforward approach in this case, thanks to the perfect square trinomial. Keep practicing these techniques, and you'll become a quadratic function master in no time! Keep your mind sharp, and happy solving!