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Magazine Chapter 3: Problem 82
Find the zeros of the function algebraically. Give exact answers. $$f(x)=3 x^{2}+5 x+1$$
Short Answer
Expert verified
The zeros are \(x = \frac{-5 + \sqrt{13}}{6}\) and \(x = \frac{-5 - \sqrt{13}}{6}\).
Step by step solution
01
- Understand the Problem
We need to find the zeros of the quadratic function \(f(x)=3x^{2}+5x+1\). This means we need to find the values of \(x\) for which \(f(x) = 0\).
02
- Set Function Equal to Zero
Set the function equal to zero: \[ 3x^{2} + 5x + 1 = 0 \]
03
- Identify Coefficients
Identify the coefficients from the quadratic equation \(ax^2 + bx + c = 0\). Here, \(a = 3\), \(b = 5\), and \(c = 1\).
04
- Use Quadratic Formula
Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the zeros.
05
- Calculate the Discriminant
Calculate the discriminant \(b^2 - 4ac\). For this equation, it is \(5^2 - 4(3)(1) = 25 - 12 = 13\).
06
- Substitute Values into Quadratic Formula
Substitute the values into the quadratic formula: \[ x = \frac{-5 \pm \sqrt{13}}{6} \]
07
- Simplify the Answers
Simplify the values to get the exact zeros: \[ x = \frac{-5 + \sqrt{13}}{6} \quad \text{and} \quad x = \frac{-5 - \sqrt{13}}{6} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). This formula helps us find the exact solutions, or zeros, of the quadratic function. The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation. We use the quadratic formula because it provides a straightforward way to find the roots, especially when the equation doesn't factor easily. To use the quadratic formula effectively:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation. We use the quadratic formula because it provides a straightforward way to find the roots, especially when the equation doesn't factor easily. To use the quadratic formula effectively:
- Identify the coefficients \(a\), \(b\), and \(c\) from your quadratic equation.
- Compute the discriminant \(b^2 - 4ac\).
- Substitute these values into the quadratic formula to solve for \(x\).
The Importance of the Discriminant
The discriminant is a key component of the quadratic formula and significantly impacts the nature of the solutions we obtain. The discriminant is the part of the quadratic formula under the square root, \(b^2 - 4ac\). Understanding the value of the discriminant helps us determine the number and type of solutions without fully solving the equation. The discriminant can tell us:
- **Positive Discriminant:** If \(b^2 - 4ac > 0\), there are two distinct real solutions.
- **Zero Discriminant:** If \(b^2 - 4ac = 0\), there is exactly one real solution (also called a repeated or double root).
- **Negative Discriminant:** If \(b^2 - 4ac < 0\), there are no real solutions, but there are two complex solutions.
Steps to Solving Algebraic Equations
Solving quadratic equations algebraically involves a few clear steps. Following these steps ensures you understand the process and achieve the correct solutions.
- **Set the equation to zero:** Start by setting the quadratic function equal to zero, as seen in the problem where \(3x^2 + 5x + 1 = 0\).
- **Identify coefficients:** Extract the coefficients \(a\), \(b\), and \(c\). For our example, \(a=3\), \(b=5\), and \(c=1\).
- **Compute the discriminant:** Calculate the discriminant \(b^2 - 4ac\) to understand the nature of the solutions you can expect. In this example, it is \(13\).
- **Use the quadratic formula:** Substitute the values of \(a\), \(b\), and the discriminant into the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). For our example, this would be \(x = \frac{-5 \pm \sqrt{13}}{6}\).
- **Simplify the solutions:** Finally, solve for \(x\) and simplify to find the exact solutions: \[x = \frac{-5 + \sqrt{13}}{6} \quad \text{and} \quad x = \frac{-5 - \sqrt{13}}{6}\].
