Solving 2x² = 4x - 7 Using The Quadratic Formula

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In mathematics, the quadratic formula is a powerful tool for finding the solutions (or roots) of quadratic equations. A quadratic equation is a polynomial equation of the second degree, generally written in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to these equations represent the points where the parabola defined by the equation intersects the x-axis. The quadratic formula provides a direct method to calculate these solutions, regardless of the complexity of the equation. This article delves into the application of the quadratic formula to solve the equation 2x² = 4x - 7, providing a step-by-step explanation and highlighting the underlying principles.

The quadratic formula is derived from the process of completing the square and is expressed 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 in the standard form ax² + bx + c = 0.
  • The term b² - 4ac is known as the discriminant, which determines the nature of the roots.

The Discriminant and Nature of Roots

The discriminant (Δ = b² - 4ac) plays a crucial role in determining the type of solutions a quadratic equation has:

  • 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 (conjugate pairs).

Complex roots involve the imaginary unit i, where i² = -1. These roots arise when the discriminant is negative, indicating that the parabola does not intersect the x-axis.

To solve the quadratic equation 2x² = 4x - 7 using the quadratic formula, we need to follow a systematic approach:

Step 1: Rewrite the Equation in Standard Form

The first step is to rewrite the given equation in the standard form ax² + bx + c = 0. To do this, subtract 4x and add 7 to both sides of the equation:

2x² - 4x + 7 = 0

Now, we can identify the coefficients:

  • a = 2
  • b = -4
  • c = 7

Step 2: Apply the Quadratic Formula

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

x = (-(-4) ± √((-4)² - 4 * 2 * 7)) / (2 * 2)

Simplify the equation:

x = (4 ± √(16 - 56)) / 4

x = (4 ± √(-40)) / 4

Step 3: Simplify the Square Root of a Negative Number

Since the discriminant is negative (-40), the solutions will be complex numbers. We can simplify the square root of -40 as follows:

√(-40) = √(40 * -1) = √(4 * 10 * -1) = 2√10 * i

Here, i represents the imaginary unit, where i² = -1.

Step 4: Substitute the Simplified Square Root Back into the Equation

Replace √(-40) with 2√10 * i in the equation:

x = (4 ± 2√10 * i) / 4

Step 5: Simplify the Expression

Divide both terms in the numerator by 4:

x = (4 / 4) ± (2√10 * i / 4)

x = 1 ± (√10 * i / 2)

Thus, the solutions are:

x = 1 + (√10 / 2)i and x = 1 - (√10 / 2)i

Alternatively, this can be written as:

x = 1 ± (√10 / 2)i

The solutions x = 1 ± (√10 / 2)i are complex conjugates. This means they have the form a + bi and a - bi, where a and b are real numbers and i is the imaginary unit. Complex solutions arise when the discriminant (b² - 4ac) is negative, indicating that the quadratic equation has no real roots.

In the context of the quadratic equation's graph (a parabola), complex roots imply that the parabola does not intersect the x-axis. Instead, it exists entirely above or below the x-axis.

When using the quadratic formula, several common mistakes can occur. Being aware of these pitfalls can help ensure accurate solutions:

  1. Incorrectly Identifying Coefficients: Ensure that the quadratic equation is in standard form (ax² + bx + c = 0) before identifying the values of a, b, and c. A mistake here will propagate through the entire solution.
  2. Sign Errors: Pay close attention to the signs of the coefficients, especially when substituting them into the formula. A misplaced negative sign can lead to incorrect results.
  3. Miscalculating the Discriminant: The discriminant (b² - 4ac) must be calculated accurately. Ensure the correct order of operations (PEMDAS/BODMAS) is followed.
  4. Simplifying Square Roots Incorrectly: When dealing with square roots, particularly those of negative numbers, ensure they are simplified correctly. Remember that √(-1) = i.
  5. Forgetting to Simplify the Final Solution: After applying the formula, simplify the result as much as possible. This often involves reducing fractions and combining like terms.

To further solidify your understanding of the quadratic formula, consider working through additional examples. Practice solving a variety of quadratic equations, including those with real roots, repeated roots, and complex roots. This will help you become more comfortable with the process and improve your accuracy.

For example, try solving the following equations:

  1. x² - 6x + 9 = 0
  2. 3x² + 5x - 2 = 0
  3. x² + 4x + 5 = 0

By working through these examples, you can reinforce your understanding and build confidence in your problem-solving abilities.

Quadratic equations and the quadratic formula are not just abstract mathematical concepts; they have numerous applications in real-world scenarios. Understanding these applications can provide a deeper appreciation for the practical significance of this mathematical tool.

  1. Physics: Quadratic equations are used to model projectile motion. For example, the trajectory of a ball thrown into the air can be described by a quadratic equation, allowing us to calculate the ball's height and range.
  2. Engineering: Engineers use quadratic equations in various applications, such as designing bridges and other structures. They help determine the optimal dimensions and materials to ensure structural integrity.
  3. Finance: Quadratic equations can be used to model financial situations, such as calculating compound interest or determining the break-even point for a business.
  4. Computer Graphics: In computer graphics, quadratic equations are used to create curves and surfaces. They are essential for rendering realistic images and animations.
  5. Optimization Problems: Quadratic equations are used to solve optimization problems, where the goal is to find the maximum or minimum value of a function. These problems arise in various fields, such as economics and logistics.

By recognizing the widespread applications of quadratic equations, we can see their importance in both theoretical and practical contexts.

The quadratic formula is an indispensable tool for solving quadratic equations. By following a systematic approach and understanding the underlying principles, you can confidently tackle a wide range of quadratic equations. In the given problem, we successfully used the quadratic formula to find the complex solutions for the equation 2x² = 4x - 7. Remember to practice regularly and be mindful of common mistakes to enhance your proficiency. Mastering the quadratic formula not only improves your mathematical skills but also provides a foundation for more advanced mathematical concepts and real-world applications.