Solving Quadratic Equations Using The Quadratic Formula A Step-by-Step Guide

In the realm of mathematics, quadratic equations hold a prominent position, frequently encountered in various fields, from physics to engineering. These equations, characterized by their highest power of the variable being two, can be solved using a variety of methods, among which the quadratic formula stands out as a universally applicable technique. This article delves into the intricacies of solving quadratic equations using the quadratic formula, providing a comprehensive guide for students and enthusiasts alike. We will specifically address the equation $4x^2 - 3x + 9 = 2x + 1$, demonstrating the step-by-step application of the formula to arrive at the solutions.

Understanding Quadratic Equations

Before we delve into the specifics of the quadratic formula, let's first establish a clear understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is two. The general form of a quadratic equation is expressed as:

$ax^2 + bx + c = 0$

where 'a', 'b', and 'c' are constants, and 'x' represents the variable we aim to solve for. The coefficient 'a' cannot be zero, as this would reduce the equation to a linear form.

Quadratic equations arise in numerous contexts, modeling phenomena such as projectile motion, the trajectory of a ball thrown in the air, or the shape of a suspension bridge cable. Their versatility and prevalence underscore the importance of mastering techniques for solving them.

The Quadratic Formula: A Universal Solution

The quadratic formula provides a direct method for finding the solutions (also known as roots) of any quadratic equation. It is a powerful tool that bypasses the need for factoring or completing the square, especially when dealing with complex or non-factorable equations. The formula is stated as follows:

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

This formula yields two possible solutions for 'x', denoted by the '$\pm$' symbol, which signifies both addition and subtraction. These solutions may be real or complex numbers, depending on the value of the discriminant, which is the expression under the square root, $b^2 - 4ac$.

Decoding the Discriminant: Nature of the Roots

The discriminant, $b^2 - 4ac$, plays a crucial role in determining the nature of the roots of the quadratic equation. It provides valuable insights into whether the solutions are real and distinct, real and equal, or complex conjugates.

1. If $b^2 - 4ac > 0$: The equation has two distinct real roots. This means there are two different real numbers that satisfy the equation.
2. If $b^2 - 4ac = 0$: The equation has one real root (or two equal real roots). This indicates that the parabola represented by the quadratic equation touches the x-axis at exactly one point.
3. If $b^2 - 4ac < 0$: The equation has two complex conjugate roots. This implies that the solutions involve imaginary numbers, and the parabola does not intersect the x-axis.

Understanding the discriminant allows us to predict the type of solutions we will obtain even before applying the quadratic formula, saving time and providing a deeper understanding of the equation's behavior.

Step-by-Step Solution: Applying the Quadratic Formula

Now, let's apply the quadratic formula to solve the given equation: $4x^2 - 3x + 9 = 2x + 1$.

Step 1: Rearrange the Equation into Standard Form

The first step is to rearrange the equation into the standard quadratic form, $ax^2 + bx + c = 0$. To do this, we need to move all terms to one side of the equation:

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

Combining like terms, we get:

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

Now, we can clearly identify the coefficients: $a = 4$, $b = -5$, and $c = 8$.

Step 2: Identify the Coefficients

As we've already done in the previous step, we identify the coefficients 'a', 'b', and 'c' from the standard form of the equation. In our case:

• a = 4
• b = -5
• c = 8

These coefficients are the key ingredients for the quadratic formula.

Step 3: Substitute the Coefficients into the Quadratic Formula

Next, we substitute the values of 'a', 'b', and 'c' into the quadratic formula:

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

This step involves careful substitution, ensuring that each coefficient is placed correctly in the formula.

Step 4: Simplify the Expression

Now, we simplify the expression step-by-step:

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

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

Notice that the discriminant is negative (-103), indicating that the solutions will be complex numbers.

Step 5: Express the Solution in Complex Form

Since the discriminant is negative, we introduce the imaginary unit, 'i', where $i = \sqrt{-1}$. We can rewrite the expression as:

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

This gives us two complex solutions:

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

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

These are the complex conjugate roots of the quadratic equation.

Final Answer: The Solutions to the Equation

Therefore, the solutions to the quadratic equation $4x^2 - 3x + 9 = 2x + 1$ are:

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

This corresponds to option C in the given choices.

Conclusion: Mastering the Quadratic Formula

The quadratic formula is an indispensable tool in the mathematician's arsenal, providing a reliable method for solving quadratic equations of any form. By understanding the formula, the discriminant, and the steps involved in applying it, one can confidently tackle a wide range of problems. From rearranging equations into standard form to simplifying complex expressions, each step contributes to the final solution. As demonstrated in this article, the quadratic formula not only provides the answers but also sheds light on the nature of the roots, whether they are real, complex, or repeated.

By mastering the quadratic formula, students and enthusiasts alike can unlock a deeper understanding of quadratic equations and their applications in various fields. Whether it's calculating projectile trajectories or modeling physical phenomena, the quadratic formula remains a cornerstone of mathematical problem-solving.

Understanding the Problem: Solving Quadratic Equations

In this comprehensive guide, we will tackle the quadratic equation $4x^2 - 3x + 9 = 2x + 1$ using the quadratic formula. Our main goal is to find the values of $x$ that satisfy this equation. Solving quadratic equations is a fundamental concept in algebra, with widespread applications in fields such as physics, engineering, and computer science. This article provides a step-by-step solution, ensuring clarity and understanding for students and math enthusiasts alike.

Why Use the Quadratic Formula?

The quadratic formula is a powerful tool for solving quadratic equations, especially those that are difficult or impossible to factor. The formula is derived from the method of completing the square and can be applied to any quadratic equation in the standard form: $ax^2 + bx + c = 0$. This method provides a direct way to find the solutions (also known as roots) of the equation, regardless of whether the roots are real or complex. In cases where factoring is not straightforward, the quadratic formula provides a reliable and efficient approach.

Step 1: Transforming the Equation into Standard Form

The first crucial step in solving any quadratic equation using the quadratic formula is to transform it into the standard form: $ax^2 + bx + c = 0$. This form allows us to easily identify the coefficients $a$, $b$, and $c$, which are necessary for applying the formula. Our given equation is $4x^2 - 3x + 9 = 2x + 1$. To transform it, we need to move all terms to one side of the equation, leaving zero on the other side. We start by subtracting $2x$ and $1$ from both sides:

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

Next, we combine like terms:

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

This simplifies to:

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

Now, our equation is in the standard form, and we can clearly identify the coefficients: $a = 4$, $b = -5$, and $c = 8$.

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

As we've transformed the equation into the standard form $ax^2 + bx + c = 0$, we can now easily identify the coefficients. These coefficients are essential for the next step, which involves plugging them into the quadratic formula. From the equation $4x^2 - 5x + 8 = 0$, we have:

• $a = 4$
• $b = -5$
• $c = 8$

It's crucial to correctly identify these coefficients, as any mistake here will lead to an incorrect solution. Pay special attention to the signs (positive or negative) of each coefficient.

Step 3: Applying the Quadratic Formula

The quadratic formula is given by:

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

This formula provides the solutions for $x$ in any quadratic equation of the form $ax^2 + bx + c = 0$. We will now substitute the coefficients we identified in the previous step ($a = 4$, $b = -5$, and $c = 8$) into the formula:

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

This substitution is a critical step, and it's important to be meticulous to avoid errors. Ensure each coefficient is placed in the correct spot within the formula.

Step 4: Simplifying the Expression: A Step-by-Step Approach

After substituting the coefficients into the quadratic formula, the next step is to simplify the expression. This involves performing the arithmetic operations and reducing the expression to its simplest form. We begin with:

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

First, we simplify the terms inside the square root and the denominator:

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

Now, we simplify further:

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

Notice that the value inside the square root is negative (-103). This indicates that the solutions will be complex numbers, as the square root of a negative number is imaginary.

Step 5: Dealing with Complex Numbers: Introducing the Imaginary Unit

Since we have a negative number inside the square root ($\sqrt{-103}$), we need to introduce the imaginary unit, denoted by $i$, where $i = \sqrt{-1}$. We can rewrite $\sqrt{-103}$ as $\sqrt{103} \cdot \sqrt{-1}$, which is $\sqrt{103}i$. So, our expression becomes:

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

This expression gives us two complex solutions for $x$:

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

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

These are the complex conjugate roots of the quadratic equation.

The Final Solutions: Presenting the Values of x

Therefore, the solutions to the quadratic equation $4x^2 - 3x + 9 = 2x + 1$ are:

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

This matches option C in the given choices. These solutions are complex numbers, meaning they have both a real part ($\frac{5}{8}$) and an imaginary part ($\frac{\sqrt{103}}{8}i$).

Conclusion: Mastering Quadratic Equations with the Quadratic Formula

In conclusion, the quadratic formula is a powerful and versatile method for solving quadratic equations. By following a systematic approach—transforming the equation into standard form, identifying the coefficients, applying the formula, and simplifying the expression—we can find the solutions, whether they are real or complex. This article has demonstrated a step-by-step solution to the equation $4x^2 - 3x + 9 = 2x + 1$, emphasizing the importance of each step and providing clarity on how to handle complex numbers. Mastering the quadratic formula is an essential skill for anyone studying algebra and its applications.

Introduction: Quadratic Equations and Their Solutions

In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Solving quadratic equations is a fundamental skill in algebra, with applications in various fields, including physics, engineering, and finance. In this article, we will walk through the process of solving the quadratic equation $4x^2 - 3x + 9 = 2x + 1$ using the quadratic formula. This method is particularly useful when the equation cannot be easily factored.

Why the Quadratic Formula is Essential

The quadratic formula is a universal tool for finding the solutions (or roots) of any quadratic equation. While some quadratic equations can be solved by factoring or completing the square, the quadratic formula provides a reliable method that works for all quadratic equations, regardless of their complexity. The formula is given by:

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

This formula allows us to find the values of $x$ that satisfy the equation, even when the roots are complex numbers. It's an indispensable tool in algebra and beyond.

Step 1: Rearranging the Equation to Standard Form

Before we can apply the quadratic formula, we need to rearrange the given equation into the standard form $ax^2 + bx + c = 0$. Our initial equation is $4x^2 - 3x + 9 = 2x + 1$. To get it into standard form, we need to move all terms to one side of the equation. We do this by subtracting $2x$ and $1$ from both sides:

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

Now, we combine like terms:

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

This equation is now in the standard form, where we can easily identify the coefficients $a$, $b$, and $c$.

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

Now that our equation is in the standard form $4x^2 - 5x + 8 = 0$, we can identify the coefficients: $a$, $b$, and $c$. These coefficients are the key to using the quadratic formula. In this case, we have:

• $a = 4$
• $b = -5$
• $c = 8$

It's essential to identify these coefficients correctly, as any error here will lead to an incorrect solution. Pay close attention to the signs of the coefficients.

Step 3: Applying the Quadratic Formula: Substitution

With the coefficients identified, we can now apply the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. We substitute the values $a = 4$, $b = -5$, and $c = 8$ into the formula:

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

This step involves careful substitution. Ensure each value is placed correctly in the formula to avoid any errors in the calculation.

Step 4: Simplifying the Expression: Arithmetic Operations

After substituting the coefficients into the quadratic formula, we need to simplify the expression. This involves performing various arithmetic operations. We start with:

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

First, we simplify the terms inside the square root and the denominator:

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

Next, we continue simplifying:

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

Notice that we have a negative number under the square root (-103). This indicates that the solutions will be complex numbers.

Step 5: Expressing the Solutions as Complex Numbers

Since we have a negative number under the square root ($\sqrt{-103}$), we need to express the solutions as complex numbers. We introduce the imaginary unit, $i$, where $i = \sqrt{-1}$. We can rewrite $\sqrt{-103}$ as $\sqrt{103}i$. Thus, our expression becomes:

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

This gives us two complex solutions for $x$:

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

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

These are the complex conjugate roots of the quadratic equation.

Final Answer: The Complex Solutions to the Equation

Therefore, the solutions to the quadratic equation $4x^2 - 3x + 9 = 2x + 1$ are:

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

This corresponds to option C in the given multiple-choice answers. The solutions are complex numbers, indicating that the quadratic equation has no real roots.

Conclusion: Mastering the Quadratic Formula for Complex Solutions

In conclusion, we have successfully solved the quadratic equation $4x^2 - 3x + 9 = 2x + 1$ using the quadratic formula. By following a systematic approach—rearranging the equation into standard form, identifying the coefficients, applying the formula, and simplifying the expression—we found the complex solutions. This example highlights the power of the quadratic formula in solving any quadratic equation, even those with complex roots. Mastering this method is crucial for success in algebra and related fields.