Solving 15x^2 + 13x = 0 With The Quadratic Formula A Step By Step Guide

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In the realm of mathematics, quadratic equations hold a significant place, appearing in various fields ranging from physics and engineering to economics and computer science. These equations, characterized by their second-degree polynomial form, often present a challenge to solve, requiring the application of specific techniques. Among the most versatile and widely used methods for tackling quadratic equations is the quadratic formula. This powerful formula provides a direct and systematic approach to finding the solutions, or roots, of any quadratic equation, regardless of its complexity.

In this comprehensive guide, we delve into the intricacies of the quadratic formula, exploring its derivation, application, and practical implications. We will embark on a step-by-step journey, unraveling the formula's components and demonstrating its usage through a concrete example. By the end of this exploration, you will be equipped with the knowledge and skills to confidently solve quadratic equations using the quadratic formula and appreciate its significance in the broader mathematical landscape.

Before we delve into the specifics of the quadratic formula, let's take a moment to refresh our understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning that the highest power of the variable is 2. The general form of a quadratic equation is expressed as:

ax2+bx+c=0ax^2 + bx + c = 0

where a, b, and c are constants, and x represents the variable. The coefficient a cannot be zero, as this would reduce the equation to a linear form. The terms ax², bx, and c are referred to as the quadratic term, the linear term, and the constant term, respectively.

Quadratic equations can have up to two distinct solutions, also known as roots or zeros. These solutions represent the values of x that satisfy the equation, making the expression equal to zero. The nature of these solutions – whether they are real or complex, distinct or repeated – depends on the discriminant, a key component of the quadratic formula that we will explore later.

The quadratic formula is a mathematical expression that provides the solutions to any quadratic equation in the general form ax² + bx + c = 0. It is derived by completing the square on the general form of the quadratic equation and is expressed as:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula reveals that the solutions for x are determined by the coefficients a, b, and c of the quadratic equation. The presence of the plus-minus symbol (±) indicates that there are generally two solutions, corresponding to the two possible signs of the square root term. Let's break down the components of the quadratic formula to gain a deeper understanding:

  • -b: This term represents the negation of the coefficient of the linear term (bx).
  • ±: This symbol indicates that there are two possible solutions, one obtained by adding the square root term and the other by subtracting it.
  • √(b² - 4ac): This is the square root of the discriminant, a critical expression that determines the nature of the solutions.
  • b² - 4ac: This expression is known as the discriminant (often denoted as Δ). Its value determines the nature of the roots:
    • If Δ > 0, the equation has two distinct real roots.
    • If Δ = 0, the equation has one real root (a repeated root).
    • If Δ < 0, the equation has two complex roots.
  • 2a: This term represents twice the coefficient of the quadratic term (ax²).

Now, let's apply the quadratic formula to solve the given equation: 15x² + 13x = 0. This equation is already in the standard quadratic form ax² + bx + c = 0, where a = 15, b = 13, and c = 0. Let's substitute these values into the quadratic formula:

x=−13±132−4⋅15⋅02⋅15x = \frac{-13 \pm \sqrt{13^2 - 4 \cdot 15 \cdot 0}}{2 \cdot 15}

Simplifying the expression:

x=−13±16930x = \frac{-13 \pm \sqrt{169}}{30}

x=−13±1330x = \frac{-13 \pm 13}{30}

This gives us two possible solutions:

x1=−13+1330=030=0x_1 = \frac{-13 + 13}{30} = \frac{0}{30} = 0

x2=−13−1330=−2630=−1315x_2 = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15}

Therefore, the solutions to the equation 15x² + 13x = 0 are x = 0 and x = -13/15. This corresponds to option A in the given choices.

To solidify your understanding of the quadratic formula, let's outline the steps involved in applying it to solve a quadratic equation:

  1. Identify the coefficients: Begin by identifying the coefficients a, b, and c in the quadratic equation ax² + bx + c = 0.

  2. Substitute the values: Substitute the identified values of a, b, and c into the quadratic formula:

    x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

  3. Simplify the expression: Simplify the expression by performing the arithmetic operations, including evaluating the square root and reducing the fraction.

  4. Determine the solutions: Extract the two possible solutions for x by considering both the positive and negative signs of the square root term.

  5. Verify the solutions (optional): To ensure accuracy, substitute the obtained solutions back into the original quadratic equation to verify that they satisfy the equation.

Quadratic equations are not merely abstract mathematical constructs; they have a wide range of applications in various real-world scenarios. Here are a few examples:

  • Physics: Quadratic equations are used to model projectile motion, describing the trajectory of objects thrown into the air. They also appear in calculations involving energy, such as kinetic energy and potential energy.
  • Engineering: Engineers use quadratic equations to design bridges, buildings, and other structures, ensuring their stability and safety. They also play a role in electrical circuit analysis and control systems.
  • Economics: Quadratic equations can be used to model supply and demand curves, helping economists understand market behavior and predict price fluctuations. They also appear in financial calculations, such as compound interest and investment returns.
  • Computer Science: Quadratic equations are used in computer graphics to create curves and surfaces, as well as in optimization algorithms to find the best solutions to problems.

While the quadratic formula is a powerful tool, there are a few tips and tricks that can help you use it more effectively:

  • Simplify before substituting: If possible, simplify the quadratic equation before substituting the coefficients into the formula. This can reduce the complexity of the calculations.
  • Pay attention to signs: Be careful with the signs of the coefficients, especially when dealing with negative values. A mistake in the sign can lead to incorrect solutions.
  • Check the discriminant: Calculate the discriminant (b² - 4ac) before proceeding with the rest of the formula. This will tell you the nature of the roots and can save you time if the roots are complex.
  • Use a calculator: For complex calculations, don't hesitate to use a calculator. This will help you avoid errors and speed up the process.
  • Verify your solutions: Always verify your solutions by substituting them back into the original equation. This will ensure that you have found the correct answers.

The quadratic formula stands as a cornerstone in the realm of algebra, offering a systematic and reliable method for solving quadratic equations. Its versatility and applicability extend far beyond the classroom, finding practical use in various scientific, engineering, and economic disciplines. By mastering the quadratic formula and understanding its underlying principles, you equip yourself with a powerful tool for tackling a wide range of mathematical problems.

In this comprehensive guide, we have explored the derivation, application, and significance of the quadratic formula. We have dissected its components, demonstrated its usage through a step-by-step example, and highlighted its relevance in real-world scenarios. With this knowledge, you are now well-equipped to confidently solve quadratic equations and appreciate the power of this fundamental mathematical concept. Remember to practice applying the formula to various problems to further solidify your understanding and enhance your problem-solving skills. The journey of mathematical exploration is continuous, and the quadratic formula serves as a valuable stepping stone in your quest for mathematical mastery.