Solve X² - 10x - 3 = 0 With Bhaskara's Formula

Hey guys! Today, we're diving into the fascinating world of quadratic equations and tackling a specific one: x² - 10x - 3 = 0. Don't worry if it looks intimidating; we'll break it down step-by-step using good old Bhaskara's formula. Trust me, by the end of this article, you'll be a pro at solving these types of equations. So, grab your pencils, notebooks, and let's get started!

Understanding Quadratic Equations

Before we jump into Bhaskara's formula, let's make sure we're all on the same page about what a quadratic equation actually is. In mathematical terms, a quadratic equation is a polynomial equation of the second degree. This basically means it has a term with x squared (x²) as its highest power. The general form of a quadratic equation is:

ax² + bx + c = 0

Where:

• 'a', 'b', and 'c' are constants (real numbers), and
• 'x' represents the variable we're trying to solve for.

Now, why are quadratic equations so important? Well, they pop up all over the place in various fields, from physics and engineering to economics and computer science. They're used to model projectile motion, calculate areas and volumes, and even optimize financial models. So, understanding how to solve them is a valuable skill to have in your mathematical toolkit.

In our specific equation, x² - 10x - 3 = 0, we can identify the coefficients as follows:

• a = 1 (the coefficient of x²)
• b = -10 (the coefficient of x)
• c = -3 (the constant term)

With this understanding, we're now ready to unleash the power of Bhaskara's formula!

Introducing Bhaskara's Formula

Okay, guys, this is where the magic happens! Bhaskara's formula, named after the brilliant 12th-century Indian mathematician Bhāskara II, is our trusty tool for solving quadratic equations. It provides a direct method for finding the values of 'x' that satisfy the equation ax² + bx + c = 0. The formula looks like this:

x = (-b ± √(b² - 4ac)) / 2a

Yes, it might look a bit scary at first glance, but don't worry! We're going to break it down piece by piece. The formula essentially tells us that there are two possible solutions for 'x', which we get by using the plus-minus symbol (±). This symbol indicates that we need to calculate the solution twice – once with a plus sign and once with a minus sign.

The expression inside the square root, b² - 4ac, is particularly important and is known as the discriminant, often denoted by the Greek letter delta (Δ). The discriminant tells us about the nature of the roots (solutions) of the quadratic equation:

• 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 (it has two complex roots).

Before we plug our values into the main formula, let's calculate the discriminant for our equation. This will give us a heads-up on what kind of solutions to expect. For x² - 10x - 3 = 0, we have a = 1, b = -10, and c = -3. So, the discriminant is:

Δ = b² - 4ac = (-10)² - 4 * 1 * (-3) = 100 + 12 = 112

Since Δ = 112 is greater than 0, we know that our equation has two distinct real roots. Now, we're ready to apply the full Bhaskara's formula and find those roots!

Applying Bhaskara's Formula to x² - 10x - 3 = 0

Alright, let's get our hands dirty and plug the values into Bhaskara's formula. Remember, we have a = 1, b = -10, and c = -3. The formula is:

x = (-b ± √(b² - 4ac)) / 2a

Substituting the values, we get:

x = (-(-10) ± √((-10)² - 4 * 1 * (-3))) / (2 * 1)

Let's simplify this step-by-step:

x = (10 ± √(100 + 12)) / 2 x = (10 ± √112) / 2

Now, we need to simplify the square root of 112. We can break down 112 into its prime factors: 112 = 2 * 2 * 2 * 2 * 7 = 2⁴ * 7. So, √112 = √(2⁴ * 7) = 2²√7 = 4√7.

Substituting this back into our equation, we get:

x = (10 ± 4√7) / 2

We can further simplify this by dividing both terms in the numerator by 2:

x = 5 ± 2√7

So, we have two solutions for x:

• x₁ = 5 + 2√7
• x₂ = 5 - 2√7

These are the two real roots of the quadratic equation x² - 10x - 3 = 0. We've done it! We've successfully applied Bhaskara's formula to solve our equation.

Understanding the Solutions

Okay, so we've found the two solutions: x₁ = 5 + 2√7 and x₂ = 5 - 2√7. But what do these numbers actually represent? Well, in the context of a quadratic equation, the solutions (also called roots or zeros) are the points where the parabola represented by the equation intersects the x-axis. Imagine a U-shaped curve (a parabola); the roots are the x-values where that curve crosses the horizontal line.

Let's get a bit more concrete. We can approximate the values of our solutions by using a calculator to find the value of √7, which is approximately 2.646. Therefore:

• x₁ ≈ 5 + 2 * 2.646 ≈ 5 + 5.292 ≈ 10.292
• x₂ ≈ 5 - 2 * 2.646 ≈ 5 - 5.292 ≈ -0.292

So, the parabola represented by x² - 10x - 3 = 0 crosses the x-axis at approximately x = 10.292 and x = -0.292. This gives us a visual understanding of what the solutions mean.

Moreover, the solutions also tell us about the symmetry of the parabola. The axis of symmetry is a vertical line that passes through the midpoint of the two roots. In our case, the axis of symmetry is at x = (x₁ + x₂) / 2 = (10.292 - 0.292) / 2 = 5. This means the parabola is symmetric around the vertical line x = 5.

Understanding the solutions in this way not only helps us solve the equation but also provides valuable insights into the behavior of the quadratic function itself.

Alternative Methods (Optional)

While Bhaskara's formula is a powerful tool, it's not the only way to solve quadratic equations. There are a couple of other methods you might find useful, depending on the specific equation you're dealing with.

1. Factoring

Factoring involves rewriting the quadratic equation in the form (x - r₁)(x - r₂) = 0, where r₁ and r₂ are the roots. If you can factor the equation, you can easily find the roots by setting each factor equal to zero and solving for x. However, factoring isn't always straightforward, especially when the roots are irrational or complex.

For example, consider the equation x² - 5x + 6 = 0. We can factor this as (x - 2)(x - 3) = 0. Setting each factor to zero gives us x - 2 = 0 and x - 3 = 0, which leads to the solutions x = 2 and x = 3.

Unfortunately, our equation x² - 10x - 3 = 0 is not easily factorable, which is why Bhaskara's formula is a more reliable method in this case.

2. Completing the Square

Completing the square is another method that involves manipulating the equation to create a perfect square trinomial. This allows you to rewrite the equation in the form (x - h)² = k, which can then be easily solved by taking the square root of both sides.

The process involves adding and subtracting a specific constant to the equation to create the perfect square. While it's a bit more involved than Bhaskara's formula, it's a valuable technique to know and can be particularly useful in certain situations.

Both factoring and completing the square are great alternative methods, but for equations like x² - 10x - 3 = 0, Bhaskara's formula often provides the most direct and efficient solution.

Conclusion

So, there you have it, guys! We've successfully tackled the quadratic equation x² - 10x - 3 = 0 using Bhaskara's formula. We've broken down the formula step-by-step, understood the significance of the discriminant, and even explored alternative methods for solving quadratic equations. Remember, practice makes perfect, so don't hesitate to try out these techniques on other equations. Keep up the great work, and you'll be a math whiz in no time! Now you know how to apply Bhaskara's formula, and you understand how to break it down step-by-step. Remember the discriminant significance and don't hesitate to practice more quadratic equations.