Solve 7x^2 + 18x + 5 = 0 Using The Discriminant Method

Hey guys! Today, we're diving into the fascinating world of quadratic equations and tackling a classic problem: solving the equation 7x² + 18x + 5 = 0 using the discriminant method. If you've ever felt lost in a sea of formulas and coefficients, don't worry! We're going to break it down step by step, making it super easy to understand. This method is super useful, so let's get to it!

Understanding Quadratic Equations

Before we jump into the solution, let's quickly recap what quadratic equations are all about. At its core, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually 'x') is 2. The general form of a quadratic equation is:

ax² + bx + c = 0

Where:

• 'a', 'b', and 'c' are coefficients (constants), and 'a' is not equal to 0.
• 'x' is the variable we want to solve for.

In our specific equation, 7x² + 18x + 5 = 0, we can identify the coefficients as:

• a = 7
• b = 18
• c = 5

Now that we know what we're dealing with, let's move on to the discriminant method. You might be wondering, why use the discriminant? Well, it's a neat little trick that helps us determine the nature of the roots (solutions) of the quadratic equation even before we find them! Are we going to have two real solutions? One real solution? Or maybe no real solutions at all? The discriminant will tell us.

The discriminant is the part of the quadratic formula that's under the square root sign, and it's given by the formula:

Δ = b² - 4ac

Where Δ (Delta) represents the discriminant.

By calculating the discriminant, we can determine the number and type of roots:

• If Δ > 0, the equation has two distinct real roots.
• If Δ = 0, the equation has one real root (a repeated root).
• If Δ < 0, the equation has no real roots (two complex roots).

Understanding these concepts is super important, guys, because it gives us a roadmap for solving the equation. It's like knowing the destination before you start your journey – you're less likely to get lost! So, with our coefficients identified and the discriminant formula in hand, we're ready to roll up our sleeves and calculate the discriminant for our equation. Let's dive into the calculations!

Calculating the Discriminant

Okay, guys, let's get our hands dirty and calculate the discriminant for the equation 7x² + 18x + 5 = 0. As we already identified, we have:

• a = 7
• b = 18
• c = 5

The discriminant formula is:

Δ = b² - 4ac

Now, we'll substitute the values of a, b, and c into the formula:

Δ = (18)² - 4 * 7 * 5

Let's break this down step by step:

1. First, calculate (18)²: 18 multiplied by itself is 324. So, (18)² = 324.
2. Next, calculate 4 * 7 * 5: 4 times 7 is 28, and 28 times 5 is 140. So, 4 * 7 * 5 = 140.
3. Now, subtract the second result from the first: Δ = 324 - 140.
4. Finally, 324 - 140 = 184. So, Δ = 184.

Therefore, the discriminant for our equation is 184. Now, what does this tell us? Remember, we discussed how the discriminant helps us determine the nature of the roots. Since 184 > 0, this means our equation has two distinct real roots. Awesome!

Knowing this is super helpful because it sets our expectations. We're not looking for one solution or a complex solution; we know we should find two different real number solutions. This is like having a treasure map – we know there are two treasures buried somewhere, and now we just need to dig them up.

So, with the discriminant calculated and the nature of the roots determined, we're ready for the next step: using the quadratic formula to find those roots. We've laid the groundwork, and now it's time to put it all together. Let's move on and solve for x using the quadratic formula. You've got this, guys!

Applying the Quadratic Formula

Alright, guys, now that we've calculated the discriminant and know we're looking for two distinct real roots, it's time to bring in the big guns: the quadratic formula. This formula is your best friend when it comes to solving quadratic equations, and it's given by:

x = (-b ± √(b² - 4ac)) / (2a)

Notice that b² - 4ac is just our discriminant (Δ), which we've already calculated! This makes our lives so much easier. We know:

• a = 7
• b = 18
• c = 5
• Δ = 184

Now, let's plug these values into the quadratic formula:

x = (-18 ± √184) / (2 * 7)

Let's break this down into smaller steps to make it less intimidating:

1. First, simplify the denominator: 2 * 7 = 14. So, our equation now looks like:

   x = (-18 ± √184) / 14

2. Next, let's simplify the square root. √184 isn't a perfect square, but we can simplify it by finding its prime factors. 184 can be factored as 4 * 46, and since 4 is a perfect square, we can rewrite √184 as √(4 * 46) = 2√46.

   So, our equation becomes:

   x = (-18 ± 2√46) / 14

3. Now, we can simplify the entire fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us:

   x = (-9 ± √46) / 7

Great! We've simplified the quadratic formula as much as possible. Now, remember, the