Solving 10x^2=2x+3 Using The Quadratic Formula A Step-by-Step Guide

Let's dive into the world of quadratic equations and explore how to solve them using the quadratic formula. Quadratic equations are polynomial equations of the second degree, meaning they have the general form ax² + bx + c = 0, where a, b, and c are constants, and 'a' is not equal to zero. These equations pop up in various fields, from physics to engineering, making it essential to know how to tackle them. One of the most reliable methods for finding the solutions (also called roots) of a quadratic equation is the quadratic formula. This formula provides a straightforward way to determine the values of 'x' that satisfy the equation.

What is the Quadratic Formula?

The quadratic formula is a mathematical formula that provides the solutions to any quadratic equation. This formula is derived by completing the square on the general form of the quadratic equation, and it's a tool that every math enthusiast should have in their arsenal. The formula is given by:

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

Where:

• x represents the solutions or roots of the quadratic equation.
• a, b, and c are the coefficients from the quadratic equation in the standard form ax² + bx + c = 0.

The ± symbol indicates that there are generally two solutions, one obtained by adding the square root term and the other by subtracting it. The expression inside the square root, b² - 4ac, is called the discriminant. The discriminant tells us about the nature of the roots:

• If b² - 4ac > 0: There are two distinct real roots.
• If b² - 4ac = 0: There is exactly one real root (a repeated root).
• If b² - 4ac < 0: There are two complex roots.

Why Use the Quadratic Formula?

The quadratic formula is a powerful tool because it works for any quadratic equation, regardless of whether it can be factored easily. Factoring is a great method when it's applicable, but many quadratic equations have roots that are not integers or simple fractions, making factoring difficult or impossible. The quadratic formula provides a reliable and systematic way to find the roots in all cases. So, whether you're dealing with a simple equation or a complex one, the quadratic formula has got your back.

Step-by-Step Guide: Solving with the Quadratic Formula

Now, let's break down the process of using the quadratic formula with a step-by-step guide. We'll use the example equation 10x² = 2x + 3 to illustrate each step. Solving quadratic equations might seem daunting, but with a clear methodology, it becomes a breeze. So, let's get started!

Step 1: Rewrite the Equation in Standard Form

Before you can apply the quadratic formula, you need to make sure your equation is in the standard form: ax² + bx + c = 0. This means rearranging the terms so that all the terms are on one side of the equation, and the other side is zero. For our example equation, 10x² = 2x + 3, we need to subtract 2x and 3 from both sides. Rewriting equations is a crucial first step, ensuring we correctly identify the coefficients a, b, and c.

10x² = 2x + 3

10x² - 2x - 3 = 0

Now, our equation is in standard form. This step is crucial because the values of a, b, and c are directly taken from this standard form. If you skip this step or do it incorrectly, you'll end up with the wrong values, leading to incorrect solutions. Think of it as laying the foundation for your solution—a strong foundation ensures a solid final result.

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

Once your equation is in standard form, identify the coefficients a, b, and c. These are the numerical values that multiply the x² term, the x term, and the constant term, respectively. In our equation, 10x² - 2x - 3 = 0:

• a = 10 (the coefficient of x²)
• b = -2 (the coefficient of x)
• c = -3 (the constant term)

Carefully note the signs of the coefficients. A negative sign is just as important as the number itself. Getting these values right is essential because they will be plugged directly into the quadratic formula. Think of a, b, and c as the key ingredients in your formula; if you measure them incorrectly, the final result won't be what you expect. So, double-check your values to ensure accuracy.

Step 3: Plug the Coefficients into the Quadratic Formula

Now comes the heart of the process: plugging the values of a, b, and c into the quadratic formula:

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

Substitute a = 10, b = -2, and c = -3 into the formula:

$$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(10)(-3)}}{2(10)}$$

This step is where the formula comes to life. It might look a bit intimidating with all the numbers and symbols, but it's simply a matter of careful substitution. Make sure you replace each variable with its corresponding value, paying close attention to signs and parentheses. A little bit of extra care here can save you from making common mistakes. Think of it as assembling a puzzle; each piece (the coefficients) needs to fit correctly in its place (the formula).

Step 4: Simplify the Expression

After substituting the values, the next step is to simplify the expression. This involves performing the arithmetic operations inside the formula. Let's break it down:

$$x = \frac{2 \pm \sqrt{4 + 120}}{20}$$

$$x = \frac{2 \pm \sqrt{124}}{20}$$

Continue simplifying the square root. We can simplify √124 by factoring out the largest perfect square, which is 4:

$$x = \frac{2 \pm \sqrt{4 \cdot 31}}{20}$$

$$x = \frac{2 \pm 2\sqrt{31}}{20}$$

Now, we can simplify the entire expression by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

$$x = \frac{1 \pm \sqrt{31}}{10}$$

Simplifying the expression is a critical step to get to the most accurate and manageable form of the solution. It involves carefully following the order of operations and looking for opportunities to reduce the expression. Think of it as refining your solution; removing any unnecessary clutter to reveal the core answer. Accurate simplification not only makes the solution easier to understand but also reduces the chances of making errors in the final step.

Step 5: Find the Two Solutions

The ± sign in the formula indicates that there are two solutions: one where we add the square root term and one where we subtract it. So, we have:

$$x_1 = \frac{1 + \sqrt{31}}{10}$$

$$x_2 = \frac{1 - \sqrt{31}}{10}$$

These are the two solutions to the quadratic equation 10x² = 2x + 3. You can leave the solutions in this exact form, or you can use a calculator to find approximate decimal values if needed.

Finding the two solutions is the final step in solving the quadratic equation. It's the culmination of all the previous steps, where we separate the ± sign to reveal the two distinct roots. Think of it as the final unveiling of the answer; the two solutions that satisfy the original equation. Whether you leave the solutions in exact form or approximate them as decimals depends on the context of the problem and what's being asked. In some cases, the exact form is preferred for its precision, while in other cases, decimal approximations might be more practical.

Common Mistakes to Avoid

When using the quadratic formula, there are some common mistakes that students often make. Being aware of these pitfalls can help you avoid them. Avoiding common mistakes is key to mastering any mathematical technique.

Mistake 1: Incorrectly Identifying a, b, and c

One of the most frequent errors is misidentifying the coefficients a, b, and c. This usually happens when the equation is not in standard form or when the signs are overlooked. Coefficient identification is the foundation of using the quadratic formula correctly.

How to Avoid:

• Always rewrite the equation in the standard form ax² + bx + c = 0 before identifying the coefficients.
• Pay close attention to the signs of the coefficients. For example, in the equation 2x² - 3x + 1 = 0, a = 2, b = -3, and c = 1.
• Double-check your values to ensure accuracy.

Mistake 2: Sign Errors

Sign errors are another common issue, particularly when dealing with negative coefficients. The quadratic formula involves several negative signs, so it's easy to make a mistake if you're not careful. Sign errors can completely change the outcome of the solution.

How to Avoid:

• Use parentheses when substituting values into the formula, especially for negative numbers. For example, if b = -5, write -(-5) instead of just -5.
• Double-check each sign as you simplify the expression.
• Take your time and be meticulous.

Mistake 3: Incorrect Simplification

Simplifying the expression under the square root and the entire formula can be tricky. Errors in arithmetic or in simplifying radicals can lead to incorrect solutions. Simplification errors are often the result of rushing through the steps.

How to Avoid:

• Follow the order of operations (PEMDAS/BODMAS) carefully.
• Simplify the expression under the square root first.
• Look for perfect square factors to simplify radicals.
• If possible, simplify the entire expression by dividing out common factors.

Mistake 4: Forgetting the ± Sign

The ± sign in the quadratic formula indicates that there are two solutions. Forgetting this sign means you'll only find one solution instead of both. Omitting the ± sign is a critical error that leads to an incomplete answer.

How to Avoid:

• Always remember that the quadratic formula gives two solutions.
• Write out both solutions separately after simplifying the expression.
• Double-check your work to make sure you haven't missed one of the solutions.

Mistake 5: Calculator Errors

Using a calculator to evaluate the solutions can introduce errors if you're not careful with the input. Incorrectly entering the formula or the values can lead to wrong answers. Calculator errors are often due to hasty or incorrect input.

How to Avoid:

• Enter the formula carefully, paying attention to parentheses and signs.
• Double-check your input before pressing the equals button.
• If possible, break the calculation into smaller steps to reduce the chance of error.
• Use a scientific calculator that follows the correct order of operations.

By being aware of these common mistakes and taking steps to avoid them, you can increase your accuracy and confidence in using the quadratic formula. Think of each step as a checkpoint; by carefully reviewing your work at each stage, you can ensure a correct and complete solution.

Real-World Applications of Quadratic Equations

Quadratic equations aren't just abstract mathematical concepts; they have numerous real-world applications. From physics to engineering, these equations help us model and solve various problems. Understanding the real-world relevance of quadratic equations makes learning them even more meaningful.

Physics

In physics, quadratic equations are used to describe projectile motion. For example, the height of a projectile (like a ball thrown into the air) as a function of time can be modeled by a quadratic equation. The quadratic formula can then be used to find the time when the projectile hits the ground or reaches its maximum height. This is crucial in fields like sports science and ballistics.

• Example: Calculating the trajectory of a ball in sports or the path of a rocket.

Engineering

Engineers use quadratic equations in various applications, such as designing bridges and buildings. The shape of a parabolic arch, commonly used in bridge construction, is described by a quadratic equation. Engineers use these equations to ensure the stability and safety of structures. Engineering applications of quadratic equations are vital for structural integrity.

• Example: Determining the dimensions of a parabolic arch bridge or calculating stress and strain in structural components.

Computer Graphics

Quadratic equations play a significant role in computer graphics and video game development. They are used to create curves and surfaces, which are essential for rendering realistic images and animations. Computer graphics rely heavily on mathematical equations like quadratics for visual effects.

• Example: Designing the curves of a character's face or creating smooth transitions in animations.

Economics

In economics, quadratic equations can be used to model cost, revenue, and profit functions. Businesses can use these models to determine the optimal production level that maximizes profit. Economic modeling often involves the use of quadratic functions to find optimal solutions.

• Example: Finding the break-even point for a business or determining the price that maximizes revenue.

Financial Mathematics

Quadratic equations are also used in financial mathematics, particularly in calculating compound interest and loan payments. They help in determining the time it takes for an investment to reach a certain value or the interest rate required to pay off a loan within a specific period. Financial applications of quadratic equations are important for investment and loan calculations.

• Example: Calculating the time to double an investment or determining the interest rate on a loan.

Optimization Problems

Many optimization problems in mathematics and other fields can be solved using quadratic equations. These problems involve finding the maximum or minimum value of a function, which often occurs at the vertex of a parabola (the graph of a quadratic equation). Optimization problems frequently leverage quadratic equations to find extreme values.

• Example: Maximizing the area of a rectangular garden with a fixed perimeter or minimizing the cost of production.

By understanding these real-world applications, you can see that quadratic equations are not just abstract concepts but powerful tools for solving practical problems in various fields. This knowledge can make learning and mastering these equations even more rewarding.

Conclusion

The quadratic formula is a versatile and essential tool for solving quadratic equations. By following the steps outlined in this guide and avoiding common mistakes, you can confidently tackle any quadratic equation that comes your way. Remember to rewrite the equation in standard form, identify the coefficients, plug them into the formula, simplify the expression, and find the two solutions. Whether you're a student, an engineer, or just someone who loves math, mastering the quadratic formula will undoubtedly come in handy. So, keep practicing, and you'll become a pro at solving quadratic equations in no time!