What Are The Roots Of Y X2 3x 10

What Are the Roots of y = x² + 3x - 10? A Comprehensive Guide to Quadratic Equations

Finding the roots of a quadratic equation is a fundamental concept in algebra. This article delves into the process of determining the roots of the equation y = x² + 3x - 10, exploring various methods and providing a comprehensive understanding of the underlying principles. We'll cover factoring, the quadratic formula, and graphical representation, ensuring you grasp this vital mathematical concept thoroughly.

Understanding Quadratic Equations and Their Roots

A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually 'x') is 2. The general form is expressed as:

ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero.

The roots (or zeros) of a quadratic equation are the values of 'x' that satisfy the equation, making it equal to zero. Graphically, these roots represent the x-intercepts of the parabola representing the quadratic function.

Method 1: Factoring the Quadratic Equation

Factoring is a straightforward method to find the roots if the quadratic equation can be easily factored. This method involves expressing the quadratic expression as a product of two linear expressions.

Let's factor our equation, y = x² + 3x - 10:

We need to find two numbers that add up to 3 (the coefficient of 'x') and multiply to -10 (the constant term). These numbers are 5 and -2. Therefore, we can factor the equation as follows:

y = (x + 5)(x - 2)

To find the roots, we set y = 0:

(x + 5)(x - 2) = 0

This equation is satisfied if either (x + 5) = 0 or (x - 2) = 0. Solving for x gives us the roots:

• x = -5
• x = 2

Therefore, the roots of the equation y = x² + 3x - 10 are -5 and 2.

Method 2: Using the Quadratic Formula

The quadratic formula provides a universal method for finding the roots of any quadratic equation, regardless of whether it can be easily factored. The formula is derived from completing the square and is expressed as:

x = [-b ± √(b² - 4ac)] / 2a

In our equation, y = x² + 3x - 10, we have a = 1, b = 3, and c = -10. Substituting these values into the quadratic formula gives:

x = [-3 ± √(3² - 4 * 1 * -10)] / (2 * 1) x = [-3 ± √(9 + 40)] / 2 x = [-3 ± √49] / 2 x = [-3 ± 7] / 2

This gives us two solutions:

• x = (-3 + 7) / 2 = 4 / 2 = 2
• x = (-3 - 7) / 2 = -10 / 2 = -5

Again, we find that the roots of the equation are 2 and -5. The quadratic formula confirms the results obtained through factoring.

Method 3: Graphical Representation

A graphical approach provides a visual understanding of the roots. Plotting the quadratic function y = x² + 3x - 10 reveals a parabola. The x-intercepts of this parabola correspond to the roots of the equation.

While manually plotting points can be time-consuming, utilizing graphing calculators or software like Desmos or GeoGebra provides an efficient way to visualize the parabola. By observing the points where the parabola intersects the x-axis, you can visually identify the roots as x = -5 and x = 2. This visual representation reinforces the previously calculated roots.

The Discriminant: Understanding the Nature of Roots

The expression inside the square root in the quadratic formula, b² - 4ac, is called the discriminant. The discriminant provides valuable information about the nature of the roots:

• If b² - 4ac > 0: The equation has two distinct real roots. This is the case for our equation y = x² + 3x - 10, as the discriminant is 49 (7²), which is positive.
• If b² - 4ac = 0: The equation has one real root (a repeated root). The parabola touches the x-axis at a single point.
• If b² - 4ac < 0: The equation has no real roots. The parabola does not intersect the x-axis; the roots are complex conjugates.

Applications of Finding Roots

Finding the roots of quadratic equations has numerous applications across various fields:

• Physics: Calculating projectile motion, determining the trajectory of objects under the influence of gravity.
• Engineering: Designing structures, analyzing stresses and strains in materials.
• Economics: Modeling supply and demand curves, analyzing market equilibrium.
• Computer Science: Developing algorithms, solving optimization problems.
• Mathematics: Solving more complex equations, exploring higher-degree polynomials.

Solving Related Quadratic Equations

Understanding the methods used for y = x² + 3x - 10 allows you to effectively solve similar quadratic equations. Let's consider some examples:

Example 1: y = x² - 4x + 4

This equation can be factored as (x - 2)(x - 2) = 0, yielding a repeated root of x = 2. The discriminant is 0, confirming the single real root.

Example 2: y = x² + x + 1

This equation cannot be easily factored. Using the quadratic formula, we get a negative discriminant, indicating no real roots; the roots are complex numbers.

Example 3: y = 2x² - 5x + 2

This equation can be factored as (2x - 1)(x - 2) = 0, giving roots x = 1/2 and x = 2.

Conclusion: Mastering Quadratic Equations

Finding the roots of a quadratic equation is a fundamental skill in algebra with far-reaching applications. This article has comprehensively explored various methods, including factoring, the quadratic formula, and graphical representation, to solve the equation y = x² + 3x - 10. Understanding these methods, along with the concept of the discriminant, allows you to solve a wide range of quadratic equations and apply this knowledge to diverse problems across numerous disciplines. Remember to practice regularly and explore different approaches to strengthen your understanding and problem-solving abilities. The more you practice, the more intuitive the process of finding the roots of quadratic equations will become.