Solving 2x² + 6x + 13 = 0 Using The Quadratic Formula A Step-by-Step Guide

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The quadratic formula is a powerful tool for finding the solutions (also called roots) of quadratic equations. A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. This comprehensive guide provides a step-by-step approach to solving quadratic equations using the quadratic formula, with a detailed example to illustrate the process. If you've ever struggled with factoring or completing the square, the quadratic formula offers a reliable alternative to find the solutions to any quadratic equation.

The Quadratic Formula: Unveiling the Core Concept

The quadratic formula itself is derived from the method of completing the square and provides a direct way to calculate the roots of a quadratic equation. The formula is as follows:

x = (-b ± √(b² - 4ac)) / 2a

Where:

  • x represents the solutions or roots of the quadratic equation.
  • a, b, and c are the coefficients of the quadratic equation 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 b² - 4ac under the square root is known as the discriminant. The discriminant plays a crucial role in determining the nature of the solutions.

Understanding the Discriminant: Nature of Solutions

The discriminant, b² - 4ac, provides valuable information about the nature of the solutions to the quadratic equation without actually solving the equation. Here's how:

  • If b² - 4ac > 0: The equation has two distinct real solutions. This means the parabola represented by the quadratic equation intersects the x-axis at two different points.
  • If b² - 4ac = 0: The equation has exactly one real solution (a repeated root). This means the parabola touches the x-axis at exactly one point.
  • If b² - 4ac < 0: The equation has no real solutions. Instead, it has two complex solutions. This means the parabola does not intersect the x-axis.

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

Here's a detailed step-by-step guide on how to solve quadratic equations using the quadratic formula:

Step 1: Identify the Coefficients

  • Begin by writing the quadratic equation in the standard form: ax² + bx + c = 0. It's crucial to ensure that the equation is set equal to zero before identifying the coefficients.
  • Carefully identify the values of the coefficients a, b, and c. Pay close attention to the signs (positive or negative) of these coefficients, as they are critical for accurate calculations. For example, in the equation 3x² - 5x + 2 = 0, a = 3, b = -5, and c = 2.

Step 2: Calculate the Discriminant

  • The discriminant, denoted as Δ (Delta), is calculated using the formula: Δ = b² - 4ac. The discriminant will tell us the nature of the solution.
  • Substitute the values of a, b, and c that you identified in Step 1 into this formula. Be mindful of the order of operations (PEMDAS/BODMAS) when performing the calculation. A common mistake is to incorrectly handle the negative signs, so double-check your work. For instance, if b is negative, remember that squaring a negative number results in a positive number.

Step 3: Apply the Quadratic Formula

  • Write down the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. This formula is the heart of the method, and it's important to have it memorized or readily available. It provides the solutions x for the quadratic equation.
  • Substitute the values of a, b, and the discriminant (Δ), which you calculated in Step 2, into the quadratic formula. Ensure that you replace each variable with its corresponding value, maintaining the correct signs. This step involves careful substitution to avoid errors.

Step 4: Simplify the Expression

  • Simplify the expression obtained in Step 3. Start by simplifying the square root, if possible. If the discriminant is a perfect square (e.g., 4, 9, 16), the square root will be a whole number. If not, you may need to approximate the square root or leave it in radical form.
  • Next, perform the addition and subtraction operations in the numerator, taking into account the ± sign. This will typically result in two separate solutions for x. Remember, the ± sign indicates that there are two possible solutions: one where you add the square root term and another where you subtract it.
  • Finally, divide by the denominator (2a) to obtain the simplified solutions for x. Ensure you simplify each solution as much as possible, reducing fractions to their lowest terms.

Step 5: Check Your Solutions (Optional but Recommended)

  • To verify the accuracy of your solutions, substitute each value of x that you found back into the original quadratic equation ax² + bx + c = 0. This step is a crucial check for errors.
  • If both solutions satisfy the equation (i.e., the equation holds true when you substitute the values), then your solutions are correct. If not, you'll need to go back and carefully review your steps to identify any mistakes. This might involve rechecking your calculations, substitutions, or simplifications.

Example: Solving 2x² + 6x + 13 = 0

Let's apply the quadratic formula to solve the equation 2x² + 6x + 13 = 0. This example will walk you through each step of the process.

Step 1: Identify the Coefficients

  • The equation is already in the standard form ax² + bx + c = 0.
  • Identify the coefficients: a = 2, b = 6, and c = 13. It's important to correctly identify these values, as they will be used in the quadratic formula.

Step 2: Calculate the Discriminant

  • The discriminant is calculated using the formula Δ = b² - 4ac.
  • Substitute the values: Δ = 6² - 4 * 2 * 13 = 36 - 104 = -68. Notice that the discriminant is negative, which indicates that the equation will have complex solutions.

Step 3: Apply the Quadratic Formula

  • The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a.
  • Substitute the values: x = (-6 ± √(-68)) / (2 * 2). Here, we replace a, b, and the discriminant with their respective values.

Step 4: Simplify the Expression

  • Simplify the square root: √(-68) = √(68 * -1) = √(4 * 17 * -1) = 2i√17, where i is the imaginary unit (√-1). Since the discriminant is negative, we introduce the imaginary unit i.
  • Substitute the simplified square root back into the equation: x = (-6 ± 2i√17) / 4.
  • Divide both terms in the numerator by the denominator: x = -3/2 ± (i√17)/2. This gives us two complex solutions.

Solutions:

  • x₁ = -3/2 + (i√17)/2
  • x₂ = -3/2 - (i√17)/2

These are the two complex solutions to the quadratic equation 2x² + 6x + 13 = 0. Since the discriminant was negative, as expected, the solutions are complex.

Real-World Applications: Beyond the Textbook

Quadratic equations and the quadratic formula aren't just abstract mathematical concepts; they have numerous applications in real-world scenarios. Understanding these applications can make the learning process more engaging and relevant.

  • Physics: In physics, quadratic equations are frequently used to describe the motion of projectiles. For example, the trajectory of a ball thrown into the air can be modeled using a quadratic equation. The quadratic formula can then be used to determine the time it takes for the ball to reach a certain height or to hit the ground. Similarly, in electrical circuits, quadratic equations are used to analyze circuits containing resistors, inductors, and capacitors.
  • Engineering: Engineers use quadratic equations in various applications, such as designing bridges and buildings. For instance, the shape of a parabolic bridge arch can be described by a quadratic equation. The quadratic formula can help engineers determine the dimensions and stability of such structures. In mechanical engineering, quadratic equations are used to analyze the vibrations and oscillations of mechanical systems.
  • Finance: In finance, quadratic equations are used in investment analysis and financial modeling. For example, the breakeven point for a business venture can be calculated using a quadratic equation. The quadratic formula can also be used to model the growth of investments over time, especially when considering compound interest. Furthermore, it can be applied in portfolio optimization to determine the optimal allocation of assets.
  • Computer Graphics: Quadratic equations play a crucial role in computer graphics, particularly in creating curves and surfaces. Bézier curves, which are fundamental in computer-aided design (CAD) and animation, are defined using quadratic (and cubic) polynomials. The quadratic formula can be used to determine points along these curves and to render them on a screen. These curves are essential for creating smooth and realistic images and animations.

Common Mistakes to Avoid: Ensuring Accuracy

When using the quadratic formula, it's essential to be meticulous and avoid common mistakes to ensure accurate results. Here are some pitfalls to watch out for:

  • Incorrectly Identifying Coefficients: A common mistake is misidentifying the coefficients a, b, and c. This often happens when the equation isn't in the standard form (ax² + bx + c = 0) or when negative signs are overlooked. Always rearrange the equation into the standard form and carefully note the signs of the coefficients.
  • Sign Errors: Sign errors are a frequent source of mistakes. Pay close attention to the signs of b and c when substituting them into the quadratic formula. Also, remember that squaring a negative number results in a positive number, and mishandling this can lead to incorrect calculations.
  • Order of Operations: Following the correct order of operations (PEMDAS/BODMAS) is crucial. Make sure to calculate the discriminant (b² - 4ac) correctly before applying the rest of the formula. Perform the operations inside the square root before any other operations.
  • Simplifying Radicals: If the discriminant is not a perfect square, you'll need to simplify the radical. Ensure you simplify it completely by factoring out any perfect square factors. For example, √20 should be simplified to 2√5.
  • Dividing Both Terms: When simplifying the final expression, remember to divide both terms in the numerator by the denominator. A common error is to divide only one term, leading to an incorrect solution.

Conclusion: Mastering the Quadratic Formula

The quadratic formula is an invaluable tool for solving quadratic equations. By following the step-by-step guide and understanding the underlying concepts, you can confidently tackle a wide range of quadratic equations. Remember to pay attention to the discriminant, which provides insights into the nature of the solutions. Practice is key to mastering the quadratic formula, so work through various examples to solidify your understanding. With consistent effort, you'll become proficient in using this powerful mathematical tool.