Constructing New Quadratic Equations From Given Roots

Alright guys, let's dive into some quadratic equation transformations! We're going to explore how to create new quadratic equations based on the roots of a given equation. Specifically, we'll look at scenarios where the new roots are either a multiple of or an increment from the original roots. This is a classic topic in algebra, and mastering it will seriously boost your problem-solving skills. So, grab your pencils, and let’s get started!

Understanding the Basics

Before we jump into the problems, let's quickly recap some fundamental concepts. A quadratic equation is generally expressed in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable we're trying to solve for. The solutions to this equation are called roots, often denoted as $x_1$ and $x_2$.

There are some cool relationships between the roots and the coefficients of the quadratic equation, which are super handy for these types of problems:

• Sum of the roots: $x_1 + x_2 = -\frac{b}{a}$
• Product of the roots: $x_1 \\cdot x_2 = \frac{c}{a}$

These relationships allow us to find the sum and product of the roots directly from the equation without actually solving for the roots themselves. This is extremely useful when we want to create a new equation based on these roots.

Problem Statement

We're given the quadratic equation $x^2 + 7x + 10 = 0$, and we need to construct new quadratic equations where the roots are:

1. Three times the roots of the given equation.
2. Two more than the roots of the given equation.

Let’s tackle each of these scenarios step by step.

Case 1: Roots are Three Times the Original Roots

Okay, so we want to find a new quadratic equation whose roots are three times the roots of $x^2 + 7x + 10 = 0$. Let the roots of the original equation be $x_1$ and $x_2$. Then the roots of the new equation will be $3x_1$ and $3x_2$.

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

From the given equation $x^2 + 7x + 10 = 0$, we can identify that $a = 1$, $b = 7$, and $c = 10$. Using the relationships we discussed earlier:

• Sum of the roots: $x_1 + x_2 = -\frac{b}{a} = -\frac{7}{1} = -7$
• Product of the roots: $x_1 \\cdot x_2 = \frac{c}{a} = \frac{10}{1} = 10$

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

The new roots are $3x_1$ and $3x_2$. Let's find their sum and product:

• Sum of the new roots: $3x_1 + 3x_2 = 3(x_1 + x_2) = 3(-7) = -21$
• Product of the new roots: $(3x_1)(3x_2) = 9x_1x_2 = 9(10) = 90$

Step 3: Construct the New Quadratic Equation

Now that we have the sum and product of the new roots, we can construct the new quadratic equation. If we let the new equation be $x^2 + Bx + C = 0$, then:

• Sum of the roots: $-B = -21$, so $B = 21$
• Product of the roots: $C = 90$

Therefore, the new quadratic equation is $x^2 + 21x + 90 = 0$.

So, whenever you need to find a quadratic equation whose roots are a multiple of the roots of a given equation, just find the sum and product of the original roots, multiply them by the appropriate factor, and then construct the new equation using these new values. Easy peasy!

Case 2: Roots are Two More Than the Original Roots

Now, let’s tackle the second scenario where the roots of the new equation are two more than the roots of the original equation. So, if the original roots are $x_1$ and $x_2$, the new roots will be $x_1 + 2$ and $x_2 + 2$.

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

As before, from the equation $x^2 + 7x + 10 = 0$, we have:

• Sum of the roots: $x_1 + x_2 = -7$
• Product of the roots: $x_1 \\cdot x_2 = 10$

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

This time, the new roots are $x_1 + 2$ and $x_2 + 2$. Let's calculate their sum and product:

• Sum of the new roots: $(x_1 + 2) + (x_2 + 2) = x_1 + x_2 + 4 = -7 + 4 = -3$
• Product of the new roots: $(x_1 + 2)(x_2 + 2) = x_1x_2 + 2x_1 + 2x_2 + 4 = x_1x_2 + 2(x_1 + x_2) + 4 = 10 + 2(-7) + 4 = 10 - 14 + 4 = 0$

Step 3: Construct the New Quadratic Equation

Now, let’s form the new quadratic equation $x^2 + Bx + C = 0$ using the sum and product of the new roots:

• Sum of the roots: $-B = -3$, so $B = 3$
• Product of the roots: $C = 0$

Therefore, the new quadratic equation is $x^2 + 3x + 0 = 0$, which simplifies to $x^2 + 3x = 0$.

Conclusion

So, there you have it! We've successfully constructed new quadratic equations based on transformations of the original roots. Whether you're dealing with multiples of the roots or increments to the roots, the key is to find the new sum and product and then use those values to build your new equation.

Remember, the relationships between the roots and coefficients are your best friends in these scenarios. Keep practicing, and you’ll become a pro at transforming quadratic equations in no time! And hey, if you ever get stuck, just remember these steps, and you’ll be golden. Keep up the great work, guys!