Solve The Equation For Its Roots: $ X^2 - X - 3 = 0 $Determine If The Equation Has:A. Two Roots B. One Root C. No Roots

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving the quadratic equation $x^2 - x - 3 = 0$ and determine if it has two roots, one root, or no roots.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In our equation, $a = 1$, $b = -1$, and $c = -3$.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that the solutions to the equation $ax^2 + bx + c = 0$ are given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Applying the Quadratic Formula

Now, let's apply the quadratic formula to our equation $x^2 - x - 3 = 0$. We have $a = 1$, $b = -1$, and $c = -3$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-3)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{1 \pm \sqrt{1 + 12}}{2}$$

$$x = \frac{1 \pm \sqrt{13}}{2}$$

Determining the Number of Roots

Now that we have the solutions to the equation, we can determine if it has two roots, one root, or no roots. The quadratic formula gives us two possible values for $x$, which are:

$$x = \frac{1 + \sqrt{13}}{2}$$

$$x = \frac{1 - \sqrt{13}}{2}$$

Since we have two distinct solutions, we can conclude that the equation has two roots.

Conclusion

In this article, we solved the quadratic equation $x^2 - x - 3 = 0$ using the quadratic formula. We determined that the equation has two roots, which are given by the solutions to the equation. This demonstrates the importance of the quadratic formula in solving quadratic equations and highlights the need for a clear understanding of the concept.

Real-World Applications

Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model economic systems, including supply and demand curves, and to analyze the behavior of economic variables.

Tips and Tricks

Here are some tips and tricks for solving quadratic equations:

• Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. It can be used to find the solutions to any quadratic equation.
• Check for rational roots: Before using the quadratic formula, check if the equation has any rational roots. If it does, you can use the rational root theorem to find the solutions.
• Use algebraic manipulations: Algebraic manipulations, such as factoring and completing the square, can be used to simplify the equation and make it easier to solve.

Common Mistakes

Here are some common mistakes to avoid when solving quadratic equations:

• Not using the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. Make sure to use it when solving quadratic equations.
• Not checking for rational roots: Before using the quadratic formula, check if the equation has any rational roots. If it does, you can use the rational root theorem to find the solutions.
• Not simplifying the equation: Algebraic manipulations, such as factoring and completing the square, can be used to simplify the equation and make it easier to solve.

Conclusion

In conclusion, solving quadratic equations is a crucial skill for students and professionals alike. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the solutions to any quadratic equation. By following the tips and tricks outlined in this article, you can avoid common mistakes and become proficient in solving quadratic equations.