Solving 3x^2 - 10x - 7 = 0 With The Quadratic Formula

Are you struggling with quadratic equations? You're not alone! Quadratic equations, those tricky expressions with an $x^2$ term, can seem daunting at first. But fear not! One of the most reliable methods for tackling these equations is the quadratic formula. In this comprehensive guide, we'll break down how to use the quadratic formula to solve the equation $0 = 3x^2 - 10x - 7$, step by step. So, let's dive in and conquer those quadratic equations together!

Understanding the Quadratic Formula

Before we jump into solving our specific equation, let's make sure we understand the quadratic formula itself. The quadratic formula is a powerful tool that provides the solutions (also called roots or zeros) for any quadratic equation in the standard form:

$ax^2 + bx + c = 0$

Where 'a', 'b', and 'c' are coefficients (numbers) and 'x' is the variable we're trying to solve for.

The quadratic formula is expressed as:

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

This formula might look intimidating, but it's actually quite straightforward once you understand each part. The $\pm$ symbol means that there are two possible solutions, one where you add the square root term and one where you subtract it. The expression inside the square root, $b^2 - 4ac$, is called the discriminant. The discriminant tells us about the nature of the solutions: if it's positive, there are two real solutions; if it's zero, there's one real solution; and if it's negative, there are two complex solutions.

Remember, the quadratic formula is your best friend when it comes to solving quadratic equations, especially those that are difficult or impossible to factor. It's a universal tool that always works, providing you correctly identify the coefficients a, b, and c. This formula is not just a mathematical trick; it's a fundamental concept in algebra that unlocks the solutions to a wide range of problems. From physics to engineering to economics, the quadratic formula has applications in various fields. Understanding it empowers you to solve real-world problems with confidence.

Identifying Coefficients: a, b, and c

The first step in using the quadratic formula is to correctly identify the coefficients a, b, and c from our equation: $0 = 3x^2 - 10x - 7$. This might seem simple, but it's crucial to get it right to ensure accurate results.

Let's break it down:

• a is the coefficient of the $x^2$ term. In our equation, the coefficient of $3x^2$ is 3, so $a = 3$.
• b is the coefficient of the $x$ term. In our equation, the coefficient of $-10x$ is -10, so $b = -10$. Be sure to include the negative sign!
• c is the constant term (the term without any $x$). In our equation, the constant term is -7, so $c = -7$. Again, pay attention to the sign.

It's a good practice to write down the values of a, b, and c separately before plugging them into the formula. This helps prevent errors and keeps your work organized. In our case, we have:

• $a = 3$
• $b = -10$
• $c = -7$

Once you've correctly identified these coefficients, you're well on your way to solving the equation using the quadratic formula. Remember, a small mistake in identifying these values can lead to a completely wrong answer, so double-check your work! This step is the foundation for the rest of the solution, ensuring that the subsequent calculations are based on the correct values. With the coefficients correctly identified, the next step is to carefully substitute them into the quadratic formula.

Substituting Values into the Quadratic Formula

Now that we've identified our coefficients ($a = 3$, $b = -10$, and $c = -7$), it's time to substitute them into the quadratic formula:

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

This step requires careful attention to detail to avoid errors. Let's go through it slowly and methodically.

First, we'll replace the variables in the formula with their corresponding values:

$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(3)(-7)}}{2(3)}$

Notice how we've carefully placed each value in its correct position. It's especially important to pay attention to the signs, particularly when dealing with negative numbers. The double negative in front of the -10 ($-(-10)$) will become positive, which is a common area for mistakes.

Now, let's simplify the expression step by step. First, we'll handle the negative signs and the multiplication:

$x = \frac{10 \pm \sqrt{100 + 84}}{6}$

We've simplified $-(-10)$ to 10, $(-10)^2$ to 100, and $-4(3)(-7)$ to +84. The next step is to add the numbers inside the square root:

$x = \frac{10 \pm \sqrt{184}}{6}$

We've now successfully substituted the values into the quadratic formula and simplified the expression as much as possible. The next step will involve simplifying the square root and finding the two possible solutions for x. Remember, accuracy in this substitution step is crucial, as any error here will propagate through the rest of the solution. Taking your time and double-checking your work will pay off in the end.

Simplifying the Square Root

Our equation now looks like this:

$x = \frac{10 \pm \sqrt{184}}{6}$

The next step is to simplify the square root, $\sqrt{184}$. To do this, we look for perfect square factors of 184. In other words, we want to find a number that is a perfect square (like 4, 9, 16, 25, etc.) that divides evenly into 184.

Let's find the prime factorization of 184: $184 = 2 \times 2 \times 2 \times 23 = 2^3 \times 23$. We can rewrite this as $184 = 4 \times 46$. Since 4 is a perfect square ($2^2$), we can simplify the square root.

$\sqrt{184} = \sqrt{4 \times 46} = \sqrt{4} \times \sqrt{46} = 2\sqrt{46}$

Now we can substitute this simplified square root back into our equation:

$x = \frac{10 \pm 2\sqrt{46}}{6}$

This simplification is important because it makes the expression easier to work with and allows us to potentially simplify further in the next step. Simplifying square roots often involves finding the largest perfect square factor and extracting its square root. If the number under the square root had no perfect square factors (other than 1), we wouldn't be able to simplify it further. In this case, we were able to simplify $\sqrt{184}$ to $2\sqrt{46}$, which is a significant improvement. This step demonstrates the importance of understanding how to manipulate square roots to arrive at the most simplified solution. With the square root simplified, we are one step closer to finding the final solutions for x.

Finding the Two Solutions for x

We've arrived at the equation:

$x = \frac{10 \pm 2\sqrt{46}}{6}$

Notice the $\pm$ symbol. This indicates that we have two possible solutions for x: one where we add $2\sqrt{46}$ and one where we subtract it. Let's separate these into two equations:

Solution 1: $x_1 = \frac{10 + 2\sqrt{46}}{6}$

Solution 2: $x_2 = \frac{10 - 2\sqrt{46}}{6}$

Before we calculate these solutions, we can simplify both fractions by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

Solution 1: $x_1 = \frac{5 + \sqrt{46}}{3}$

Solution 2: $x_2 = \frac{5 - \sqrt{46}}{3}$

These are the exact solutions for x. If we need approximate decimal values, we can use a calculator to find the square root of 46 (approximately 6.78) and perform the calculations:

Solution 1: $x_1 \approx \frac{5 + 6.78}{3} \approx \frac{11.78}{3} \approx 3.93$

Solution 2: $x_2 \approx \frac{5 - 6.78}{3} \approx \frac{-1.78}{3} \approx -0.59$

Therefore, the two solutions for the quadratic equation $0 = 3x^2 - 10x - 7$ are approximately 3.93 and -0.59. We've successfully used the quadratic formula to find both solutions. This step highlights the power of the quadratic formula in providing a clear and methodical way to solve quadratic equations, even when the solutions are not whole numbers or simple fractions. Remember, the $\pm$ symbol is the key to unlocking both solutions, and simplifying the fractions before calculating the decimal approximations can make the process easier and less prone to errors.

Conclusion

Congratulations! You've successfully solved the quadratic equation $0 = 3x^2 - 10x - 7$ using the quadratic formula. We've broken down each step, from identifying the coefficients to simplifying the square root and finding the two solutions. Remember, practice makes perfect! The more you use the quadratic formula, the more comfortable and confident you'll become in solving quadratic equations.

The quadratic formula is a powerful tool that you can use to solve any quadratic equation. It might seem complex at first, but by breaking it down into smaller steps, you can master this essential skill. Keep practicing, and you'll be solving quadratic equations like a pro in no time!

For further learning and practice, you can explore resources like Khan Academy's Quadratic Equations section.