Solving $4x^2 - 3x + 9 = 2x + 1$ Using The Quadratic Formula

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Understanding the Quadratic Formula

To begin, it's crucial to understand what a quadratic equation is and why the quadratic formula is so valuable. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable (usually x) is 2. They often arise in various mathematical and real-world applications, such as physics, engineering, and economics. The general form of a quadratic equation is:

$ax^2 + bx + c = 0$

where a, b, and c are constants, and a is not equal to 0. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that satisfy the equation. The quadratic formula provides a direct method for finding these solutions, regardless of whether the equation can be easily factored. The formula is given by:

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

This formula might look intimidating at first, but with practice, it becomes a straightforward tool. The $\pm$ symbol indicates that there are typically two solutions, one obtained by adding the square root term and the other by subtracting it. The expression inside the square root, $b^2 - 4ac$, is called the discriminant, and it plays a crucial role in determining the nature of the solutions. The discriminant can tell us whether the solutions are real and distinct, real and equal, or complex (involving imaginary numbers).

Step 1: Rewrite the Equation in Standard Form

The first step in using the quadratic formula is to rewrite the given equation in the standard form, $ax^2 + bx + c = 0$. This involves moving all terms to one side of the equation, leaving zero on the other side. Let's apply this to our example equation:

$4x^2 - 3x + 9 = 2x + 1$

To rewrite this in standard form, we need to subtract $2x$ and 1 from both sides:

$4x^2 - 3x + 9 - 2x - 1 = 0$

Now, combine like terms:

$4x^2 - 5x + 8 = 0$

This is now in the standard quadratic equation form, where we can identify a, b, and c. In this case:

• $a = 4$
• $b = -5$
• $c = 8$

Rewriting the equation in standard form is a crucial preparatory step, ensuring that the coefficients are correctly identified for use in the quadratic formula. Without this step, plugging the values into the formula can lead to incorrect solutions.

Step 2: Identify the Coefficients a, b, and c

Once the equation is in standard form, the next step is to accurately identify the coefficients a, b, and c. As we determined in the previous step, for the equation $4x^2 - 5x + 8 = 0$, we have:

• $a = 4$ (a is the coefficient of the $x^2$ term)
• $b = -5$ (b is the coefficient of the x term)
• $c = 8$ (c is the constant term)

Careful attention to the signs of the coefficients is essential. A common mistake is to overlook a negative sign, which can significantly alter the result. For instance, in our example, b is -5, not 5. Identifying these coefficients correctly is crucial because they are the values we will substitute into the quadratic formula.

Step 3: Apply the Quadratic Formula

Now that we have identified a, b, and c, we can substitute these values into the quadratic formula:

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

Substituting $a = 4$, $b = -5$, and $c = 8$, we get:

$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(4)(8)}}{2(4)}$

Simplifying this expression involves several steps. First, let's simplify the terms inside the formula:

$x = \frac{5 \pm \sqrt{25 - 128}}{8}$

Next, we simplify the expression under the square root:

$x = \frac{5 \pm \sqrt{-103}}{8}$

The appearance of a negative number under the square root indicates that the solutions will be complex numbers. This is because the square root of a negative number is an imaginary number.

Step 4: Simplify the Solution

Since we have a negative number under the square root, we need to express the solution in terms of the imaginary unit, i, where $i = \sqrt{-1}$. We can rewrite $\sqrt{-103}$ as $\sqrt{103} \cdot \sqrt{-1}$, which is $\sqrt{103}i$. Substituting this back into our equation, we get:

$x = \frac{5 \pm \sqrt{103}i}{8}$

This gives us two complex solutions:

$x_1 = \frac{5 + \sqrt{103}i}{8}$

$x_2 = \frac{5 - \sqrt{103}i}{8}$

These solutions are complex conjugates, meaning they have the same real part but opposite imaginary parts. This is a common occurrence when solving quadratic equations with a negative discriminant.

Step 5: Express the Solution in a + bi Form (If Applicable)

Complex numbers are often expressed in the form a + bi, where a is the real part and bi is the imaginary part. Our solutions are already in this form:

$x_1 = \frac{5}{8} + \frac{\sqrt{103}}{8}i$

$x_2 = \frac{5}{8} - \frac{\sqrt{103}}{8}i$

This representation clearly shows the real and imaginary components of the solutions. Expressing the solutions in a + bi form can be helpful for further calculations or interpretations, especially in fields like electrical engineering and quantum mechanics.

Interpreting the Discriminant

Before we conclude, it's important to understand the role of the discriminant ($b^2 - 4ac$) in determining the nature of the solutions. The discriminant provides valuable information about the roots of the quadratic equation without actually solving the equation.

1. If $b^2 - 4ac > 0$, the equation has two distinct real solutions.
2. If $b^2 - 4ac = 0$, the equation has one real solution (a repeated root).
3. If $b^2 - 4ac < 0$, the equation has two complex solutions (complex conjugates).

In our example, the discriminant is $(-5)^2 - 4(4)(8) = 25 - 128 = -103$. Since the discriminant is negative, we correctly predicted that the solutions would be complex.

Understanding the discriminant can save time and effort by providing insight into the type of solutions to expect. It is a fundamental concept in the study of quadratic equations and their properties.

Conclusion

In this comprehensive guide, we have walked through the process of using the quadratic formula to solve the equation $4x^2 - 3x + 9 = 2x + 1$. We covered each step in detail, from rewriting the equation in standard form to simplifying the complex solutions. The key takeaways include:

• Understanding the quadratic formula and its components.
• Rewriting the equation in standard form to correctly identify a, b, and c.
• Substituting the values into the formula and simplifying the expression.
• Expressing complex solutions in a + bi form.
• Interpreting the discriminant to determine the nature of the solutions.

Mastering the quadratic formula is essential for solving a wide range of quadratic equations. By following these steps and practicing regularly, you can confidently tackle even the most challenging problems. Remember, the quadratic formula is a powerful tool in your mathematical toolkit, and with consistent effort, you can become proficient in its use. The solution to the equation $4x^2 - 3x + 9 = 2x + 1$ are:

$x_1 = \frac{5 + \sqrt{103}i}{8}$

$x_2 = \frac{5 - \sqrt{103}i}{8}$

These are complex solutions, as indicated by the negative discriminant. With practice and a solid understanding of the underlying principles, you can confidently solve any quadratic equation using the quadratic formula.