Solving Quadratic Equations: A Complete Guide

by Alex Johnson 46 views

Let's dive into solving the quadratic equation 2x(3x-1)=-6(2x+1). We will explore how to find the discriminant, determine the best method for solving, and find the solutions.

Understanding the Quadratic Equation

The given equation is 2x(3x-1)=-6(2x+1). To begin, let's simplify and rewrite it in the standard quadratic form, which is ax² + bx + c = 0. Expanding the equation gives us:

6x² - 2x = -12x - 6

Now, move all terms to one side:

6x² - 2x + 12x + 6 = 0

This simplifies to:

6x² + 10x + 6 = 0

We can further simplify by dividing the entire equation by 2:

3x² + 5x + 3 = 0

Now we have the quadratic equation in standard form: 3x² + 5x + 3 = 0. Here, a = 3, b = 5, and c = 3.

Calculating the Discriminant

The discriminant is a part of the quadratic formula that helps determine the nature of the roots (solutions) of the quadratic equation. The formula for the discriminant (D) is:

D = b² - 4ac

In our equation, a = 3, b = 5, and c = 3. Plugging these values into the discriminant formula:

D = (5)² - 4(3)(3)

D = 25 - 36

D = -11

So, the discriminant is -11. The discriminant tells us about the nature of the solutions:

  • If D > 0, there are two distinct real solutions.
  • If D = 0, there is exactly one real solution (a repeated root).
  • If D < 0, there are no real solutions; instead, there are two complex solutions.

Since our discriminant is -11, which is less than 0, we know that the equation has two complex solutions.

Determining the Best Method for Solving

When it comes to solving quadratic equations, there are several methods available:

  1. Factoring: This method involves expressing the quadratic equation as a product of two binomials. It's efficient when the roots are rational numbers.

  2. Completing the Square: This method involves manipulating the equation to form a perfect square trinomial, making it easier to solve for x. It is particularly useful when the quadratic equation cannot be easily factored.

  3. Quadratic Formula: This method is universally applicable and can solve any quadratic equation, regardless of the nature of its roots. The quadratic formula is given by:

    x = (-b ± √(b² - 4ac)) / (2a)

Considering our equation 3x² + 5x + 3 = 0 and the fact that the discriminant is negative, the best method to use here is the quadratic formula. Factoring is not feasible since the roots are complex, and completing the square, while possible, is more cumbersome than directly applying the quadratic formula.

Finding the Solutions

Using the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Plugging in our values a = 3, b = 5, and c = 3:

x = (-5 ± √((-11))) / (2 * 3)

x = (-5 ± √(-11)) / 6

Since we have a negative value inside the square root, we introduce the imaginary unit i, where i = √(-1). So, √(-11) = √(11) * √(-1) = i√11.

Therefore:

x = (-5 ± i√11) / 6

This gives us two complex solutions:

x₁ = (-5 + i√11) / 6

x₂ = (-5 - i√11) / 6

So, the solutions are x = (-5 + i√11) / 6 and x = (-5 - i√11) / 6.

Table Summary

Equation Discriminant Best Method Solution(s)
2x(3x-1)=-6(2x+1) -11 Quadratic Formula (-5 ± i√11) / 6

Deep Dive into Quadratic Equations

Let's explore quadratic equations more broadly. A quadratic equation is a polynomial equation of the second degree. The general form is:

ax² + bx + c = 0

where a, b, and c are constants, and x is the variable. The solutions to a quadratic equation are also called its roots or zeros. Understanding how to find these roots is a fundamental skill in algebra.

Methods for Solving Quadratic Equations

As mentioned earlier, there are several methods to solve quadratic equations:

  1. Factoring: Factoring is the process of breaking down a quadratic expression into the product of two linear expressions. For example, x² + 5x + 6 can be factored into (x + 2)(x + 3). The roots can then be found by setting each factor equal to zero.

  2. Completing the Square: Completing the square involves transforming the quadratic equation into a perfect square trinomial plus a constant. This method is particularly useful when the quadratic equation cannot be easily factored. It involves adding and subtracting a term to both sides of the equation to create a perfect square. The general process involves manipulating the quadratic to the form (x + h)² = k, from which the solutions x = -h ± √k can be easily found. For example, to solve x² + 6x + 5 = 0 by completing the square, we would rewrite it as:

    x² + 6x = -5

    Then, add (6/2)² = 9 to both sides:

    x² + 6x + 9 = -5 + 9

    (x + 3)² = 4

    Taking the square root gives:

    x + 3 = ±2

    So, x = -3 ± 2, which means x = -1 or x = -5.

  3. Quadratic Formula: The quadratic formula is a universal method for solving quadratic equations, regardless of the nature of their roots. It is derived from the method of completing the square and is given by:

    x = (-b ± √(b² - 4ac)) / (2a)

    This formula can be used to find the roots of any quadratic equation in the form ax² + bx + c = 0. The term inside the square root, b² - 4ac, is the discriminant, which provides information about the nature of the roots.

The Discriminant Revisited

The discriminant, D = b² - 4ac, plays a crucial role in understanding the nature of the solutions. Here's a summary:

  • D > 0: Two distinct real solutions. The parabola intersects the x-axis at two different points.
  • D = 0: One real solution (a repeated root). The parabola touches the x-axis at one point.
  • D < 0: No real solutions; two complex solutions. The parabola does not intersect the x-axis.

Complex Solutions

When the discriminant is negative, the quadratic equation has complex solutions. Complex numbers have the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit, defined as i = √(-1). For instance, if the discriminant is -4, then √(-4) = √(4) * √(-1) = 2i. Complex solutions always occur in conjugate pairs, meaning if a + bi is a solution, then a - bi is also a solution.

Applications of Quadratic Equations

Quadratic equations have numerous applications in various fields, including physics, engineering, economics, and computer science. They are used to model projectile motion, optimize areas and volumes, and solve problems involving rates of change. Understanding quadratic equations is therefore essential for many practical applications.

Tips for Solving Quadratic Equations

  • Simplify: Always simplify the equation as much as possible before attempting to solve it.
  • Check: Verify your solutions by plugging them back into the original equation.
  • Choose Wisely: Select the most appropriate method based on the nature of the equation. Factoring is quick if the roots are rational, while the quadratic formula is universally applicable.
  • Practice: The more you practice, the more comfortable you will become with solving quadratic equations.

By understanding the concepts discussed in this guide, you will be well-equipped to tackle a wide range of quadratic equations and appreciate their significance in various mathematical and real-world contexts. Remember to practice regularly and apply these techniques to different problems to reinforce your understanding.

For further exploration on quadratic equations, you can visit Khan Academy's resource on Quadratic Equations.