Solving Quadratic Equations and Their Roots: A Comprehensive Guide

Understanding and solving quadratic equations is a fundamental skill in algebra. This guide provides a step-by-step walkthrough of solving two related quadratic equations and understanding the roots of each equation. We will explore the process of transforming and solving the equations, ensuring a clear and concise explanation for each step.

1. Introduction to Quadratic Equations and Their Roots

A quadratic equation is a polynomial equation of the second degree, usually expressed in the form ax2 bx c 0. Solving a quadratic equation means finding the values of the variable (in this case, x) that satisfy the equation. The roots of the quadratic equation can be found using the quadratic formula: x frac{-b pm sqrt{b^2 - 4ac}}{2a}, where a, b, and c are the coefficients of the equation.

2. Solving the First Equation

Let's start by examining the equation: x2 - 1 - 17 0.

2.1 Simplifying the Equation

First, we simplify the equation by combining the terms on the left side:

x2 - 1 - 17 0 becomes x2 - 18 0.

2.2 Finding the Roots

Now, we can find the roots using the quadratic formula. Here, a 1, b -10, and c -17. Plugging these values into the formula, we get:

x frac{-(-10) pm sqrt{(-10)^2 - 4(1)(-17)}}{2(1)} frac{10 pm sqrt{100 68}}{2} frac{10 pm sqrt{168}}{2}.

Simplifying further, we obtain:

x frac{10 pm 12.96}{2} 5 pm 6.48.

Therefore, the roots of the equation are x1 11.48 and x2 -1.48.

3. Solving the Second Equation

The second equation is given by: x2 - a x - b - 8 0.

3.1 Identifying the Coefficients

From the first equation, we know that the product of the roots a and b is ab -10 and their sum is a b 17. This allows us to rewrite the second equation in terms of the roots of the first equation.

3.2 Rewriting the Second Equation

The second equation can be rewritten as:

x2 - (a b)x ab - 8 0 x2 - 17x - 10 - 8 0.

Combining the constant terms, we get:

x2 - 17x - 18 0.

3.3 Finding the Roots of the Second Equation

Using the quadratic formula, we find the roots of this equation:

x frac{-(-17) pm sqrt{(-17)^2 - 4(1)(-18)}}{2(1)} frac{17 pm sqrt{289 72}}{2} frac{17 pm sqrt{361}}{2}.

Since sqrt{361} 19, we have:

x frac{17 pm 19}{2} 18 and 1.

Thus, the roots of the equation are x1 18 and x2 1.

4. Conclusion

In conclusion, we have explored the process of solving two related quadratic equations and determined the roots of each equation. The roots of the first equation, x2 - 1 - 17 0, are 11.48 and -1.48, while the roots of the second equation, x2 - 17x - 18 0, are 18 and 1. Mastering these concepts will significantly enhance your skills in solving quadratic equations and understanding their roots.