Quadratic Equations: Finding New Equations With Shifted Roots

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Let's dive into the fascinating world of quadratic equations and explore how to create new ones based on transformations of their roots. In this article, we'll tackle the problem of finding a new quadratic equation whose roots are shifted versions of the roots of a given quadratic equation. Specifically, we'll consider the case where the roots are shifted by a constant value. Buckle up, guys, it's gonna be a mathematical adventure!

Understanding the Basics of Quadratic Equations

Before we jump into the problem, let's quickly review some fundamental concepts about quadratic equations. A quadratic equation is a polynomial equation of the second degree, generally represented as:

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

where a, b, and c are constants, and a ≠ 0. The solutions to this equation are called roots, which we often denote as α and β. These roots can be real or complex numbers.

The relationship between the roots and the coefficients of a quadratic equation is described by Vieta's formulas. These formulas state that:

  • Sum of the roots: α + β = -b/a
  • Product of the roots: αβ = c/a

These formulas are incredibly useful for solving various problems related to quadratic equations, including the one we're about to tackle.

Problem Statement: Shifting the Roots

Okay, guys, let's get to the heart of the matter. We're given a quadratic equation:

3x2−7x−6=03x^2 - 7x - 6 = 0

We know that α and β are the roots of this equation. Our mission is to find a new quadratic equation whose roots are (α + 2) and (β + 2). In other words, we're shifting each root by 2.

Strategy: Using Vieta's Formulas and Root Transformations

To solve this problem, we'll leverage Vieta's formulas and the concept of root transformations. Here's the plan:

  1. Find the sum and product of the original roots (α and β) using Vieta's formulas on the given equation.
  2. Determine the sum and product of the new roots (α + 2 and β + 2) using the values obtained in step 1.
  3. Construct the new quadratic equation using the sum and product of the new roots.

Let's execute this plan step by step.

Step 1: Finding the Sum and Product of the Original Roots

Given the quadratic equation 3x2−7x−6=03x^2 - 7x - 6 = 0, we can identify the coefficients as a = 3, b = -7, and c = -6. Now, let's apply Vieta's formulas:

  • Sum of the roots: α + β = -b/a = -(-7)/3 = 7/3
  • Product of the roots: αβ = c/a = -6/3 = -2

So, we have α + β = 7/3 and αβ = -2. These values will be crucial in the next step.

Step 2: Determining the Sum and Product of the New Roots

Now, let's find the sum and product of the new roots, which are (α + 2) and (β + 2). Here we go:

Sum of the New Roots:

(α + 2) + (β + 2) = α + β + 4

We already know that α + β = 7/3, so:

(α + 2) + (β + 2) = 7/3 + 4 = 7/3 + 12/3 = 19/3

Product of the New Roots:

(α + 2)(β + 2) = αβ + 2α + 2β + 4 = αβ + 2(α + β) + 4

We know that αβ = -2 and α + β = 7/3, so:

(α + 2)(β + 2) = -2 + 2(7/3) + 4 = -2 + 14/3 + 4 = 2 + 14/3 = 6/3 + 14/3 = 20/3

Therefore, the sum of the new roots is 19/3, and the product of the new roots is 20/3.

Step 3: Constructing the New Quadratic Equation

Now that we have the sum and product of the new roots, we can construct the new quadratic equation. Remember that a quadratic equation can be written in terms of its roots as:

x² - (sum of roots)x + (product of roots) = 0

In our case, the sum of the new roots is 19/3, and the product of the new roots is 20/3. So, the new quadratic equation is:

x² - (19/3)x + (20/3) = 0

To get rid of the fractions, we can multiply the entire equation by 3:

3x² - 19x + 20 = 0

And there you have it, guys! This is the new quadratic equation whose roots are (α + 2) and (β + 2).

Conclusion

In this article, we successfully found the new quadratic equation with roots (α + 2) and (β + 2), given the original equation 3x2−7x−6=03x^2 - 7x - 6 = 0. We achieved this by utilizing Vieta's formulas to find the sum and product of the original roots, then calculating the sum and product of the transformed roots, and finally constructing the new quadratic equation. This problem demonstrates the power and versatility of Vieta's formulas in manipulating and transforming quadratic equations.

So, the correct answer is:

A. 3x2−19x+20=03x^2 - 19x + 20 = 0

Keep practicing, and you'll become a quadratic equation master in no time!

Additional Tips and Tricks

Here are a few extra tips to help you tackle similar problems:

  • Always start with Vieta's formulas: They are your best friend when dealing with roots and coefficients of quadratic equations.
  • Be careful with signs: Pay close attention to the signs of the coefficients and the roots. A small mistake can lead to a completely wrong answer.
  • Practice, practice, practice: The more you practice, the more comfortable you'll become with these types of problems.
  • Consider alternative methods: Sometimes, there might be alternative methods to solve the problem. Explore different approaches to find the one that works best for you.

Further Exploration

If you're interested in learning more about quadratic equations and their properties, here are some topics you might want to explore:

  • The quadratic formula: A general formula for finding the roots of any quadratic equation.
  • Discriminant: The part of the quadratic formula that determines the nature of the roots (real, complex, equal).
  • Graphing quadratic equations: Understanding the shape of a parabola and its relationship to the roots of the equation.
  • Applications of quadratic equations: Exploring real-world scenarios where quadratic equations are used.

Keep exploring, keep learning, and most importantly, have fun with math!