Solving Quadratic Equations: Types & Number Of Solutions

Hey math enthusiasts! Today, we're diving into the fascinating world of quadratic equations. Specifically, we're going to figure out how to determine the number and type of solutions for a quadratic equation. Let's tackle the equation $5x^2 - 9x - 2 = 0$. Don't worry, it's not as scary as it looks. We'll break it down step by step, using concepts that make the process a breeze. By the end of this guide, you'll be able to confidently classify the solutions of any quadratic equation you encounter. Ready to get started?

Understanding Quadratic Equations: A Quick Refresher

First things first, let's make sure we're all on the same page. A quadratic equation is an equation that can be written in the form $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. These equations are fundamental in algebra, popping up in everything from physics to engineering. In our example, $5x^2 - 9x - 2 = 0$, we can see that 'a' is 5, 'b' is -9, and 'c' is -2. The solutions to a quadratic equation are the values of 'x' that satisfy the equation, also known as the roots or zeros of the equation. These solutions can be real numbers (rational or irrational) or complex numbers. The number and type of solutions depend on the discriminant, which we'll explore in the next section.

Quadratic equations are characterized by their highest degree being 2, which results in a parabolic graph. This characteristic means that a quadratic equation can have up to two solutions. These solutions represent the points where the parabola intersects the x-axis. The nature of these intersection points, and therefore the types of solutions, depend on the coefficients of the equation and the discriminant value. Understanding the components of a quadratic equation and its graphical representation is essential for solving and analyzing these types of problems. For instance, the quadratic equation can have two real and distinct solutions, meaning the parabola intersects the x-axis at two separate points, a single repeated solution (the parabola just touches the x-axis at one point), or two complex solutions (the parabola does not intersect the x-axis at all). Each of these scenarios is determined by the specific values of a, b, and c within the equation.

We often encounter quadratic equations in various real-world applications. For example, in physics, the trajectory of a projectile (like a ball thrown in the air) is modeled using a quadratic equation. The solutions to the equation would tell us where the projectile lands. In engineering, quadratic equations help in designing structures, calculating the optimal dimensions of objects, or analyzing the stability of systems. Even in finance, they can be used to model market trends or calculate investment returns. The ability to solve and understand quadratic equations is a valuable skill in many fields.

The Discriminant: Your Key to Solution Types

Alright, let's talk about the discriminant. This is the secret weapon for determining the number and type of solutions without actually solving the equation. The discriminant is calculated using the formula: $D = b^2 - 4ac$. The value of the discriminant tells us everything we need to know about the solutions.

• If $D > 0$: The equation has two distinct real solutions.
• If $D = 0$: The equation has one repeated real solution.
• If $D < 0$: The equation has two complex solutions (no real solutions).

Back to our equation, $5x^2 - 9x - 2 = 0$, we have a = 5, b = -9, and c = -2. Let's plug these values into the discriminant formula:

$D = (-9)^2 - 4 * 5 * (-2)$ $D = 81 + 40$ $D = 121$

Since the discriminant (121) is greater than zero, we know the equation has two distinct real solutions. Furthermore, because the discriminant is a perfect square (121 = $11^2$), the solutions are rational numbers. If the discriminant had been a positive number but not a perfect square, the solutions would have been irrational.

The discriminant simplifies the process of identifying solution types. Instead of solving the entire quadratic equation using the quadratic formula, you can immediately determine the nature of the roots by calculating just the discriminant value. This approach is not only faster but also provides valuable insights into the behavior of the equation. A positive discriminant signifies that the equation's graphical representation will intersect the x-axis at two distinct points, indicating two real solutions. If the discriminant is zero, the parabola touches the x-axis at a single point, resulting in a single repeated solution. A negative discriminant suggests that the equation does not intersect the x-axis at all, implying that the solutions are complex numbers. By understanding the discriminant, you gain a foundational understanding of quadratic equations, empowering you to effectively analyze and predict the nature of solutions without the need for extensive calculations.

Solving for the Solutions (Just to be Sure!)

Although we've already determined the nature of the solutions, let's solve the equation using the quadratic formula just to confirm our findings. The quadratic formula is: x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

For our equation, $5x^2 - 9x - 2 = 0$, we have:

x = rac{-(-9) \pm \sqrt{(-9)^2 - 4 * 5 * (-2)}}{2 * 5} x = rac{9 \pm \sqrt{81 + 40}}{10} x = rac{9 \pm \sqrt{121}}{10} x = rac{9 \pm 11}{10}

This gives us two solutions:

• x_1 = rac{9 + 11}{10} = rac{20}{10} = 2
• x_2 = rac{9 - 11}{10} = rac{-2}{10} = -rac{1}{5}

Indeed, we have two distinct rational solutions: 2 and -1/5. This confirms what we deduced from the discriminant!

Using the quadratic formula, you can directly calculate the values of the roots of any quadratic equation. In this case, by substituting the values of a, b, and c into the formula, we found two real and distinct solutions: 2 and -1/5. These values validate the previous analysis based on the discriminant. The solutions represent the x-intercepts of the equation's graph, where the parabola crosses the x-axis. Knowing both the formula and the discriminant gives you a comprehensive method for solving and understanding any quadratic equation, allowing you to not only find the roots but also predict their nature without the need for advanced calculations. This combined approach is a powerful tool in solving and understanding quadratic equations.

Conclusion: Wrapping It Up

So, to recap, for the equation $5x^2 - 9x - 2 = 0$, we have two different rational-number solutions. We figured this out by calculating the discriminant (which was positive and a perfect square) and then confirmed it by using the quadratic formula. Congratulations, you've successfully navigated the world of quadratic equations! Keep practicing, and you'll become a pro in no time.

Therefore, the correct answer is C. Two different rational-number solutions.

Remember, understanding the discriminant is key. It's a quick and efficient way to determine the type and number of solutions. If the discriminant is a positive perfect square, you'll have two distinct rational solutions. If it's positive but not a perfect square, you'll have two distinct irrational solutions. If it's zero, you'll have one repeated rational solution. And if it's negative, you'll have two complex solutions. Now go forth and conquer those quadratic equations!

Tips for Future Quadratic Equation Problems

To become more proficient in solving quadratic equations, practice is key. Try solving different quadratic equations and always start by calculating the discriminant. This will give you an immediate insight into the type of solutions you can expect. Then, depending on what the problem asks, you can either solve using the quadratic formula or use other methods such as factoring, completing the square. Always check your solutions by substituting the values back into the original equation to ensure accuracy. If you are struggling, don't hesitate to seek out additional resources, such as practice problems or online tutorials. Understanding the discriminant and knowing when to apply the quadratic formula is a critical skill for any math student.

Remember to review the basic concepts frequently. Understand the components of a quadratic equation. Make sure you are familiar with the standard form, which helps organize information and set up the problem effectively. Regularly test yourself on these concepts to make sure you are improving. Consistency in your learning will build confidence and make you a more successful problem-solver.

Frequently Asked Questions

• Can a quadratic equation have no real solutions? Yes, when the discriminant is negative, the solutions are complex, and there are no real solutions.
• What if I can't remember the quadratic formula? Try to memorize it, it is a key formula in mathematics, and it will be helpful in many situations.
• What other methods can I use to solve quadratic equations? Factoring and completing the square are other methods to solve quadratic equations.