Solving Quadratic Equations: Finding Roots With The Quadratic Formula

by ADMIN 70 views

Hey guys! Today, we're diving into the world of quadratic equations and tackling a classic problem: finding the roots of an equation using the quadratic formula. Specifically, we'll be working with the equation x2+16=10xx^2 + 16 = 10x. Don't worry, it sounds more intimidating than it actually is. We'll break it down step-by-step, so you'll be a quadratic equation whiz in no time!

Understanding Quadratic Equations

Before we jump into the quadratic formula, let's make sure we're all on the same page about what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree. That basically means it has a term with x2x^2 in it. The general form of a quadratic equation is:

ax2+bx+c=0ax^2 + bx + c = 0

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The roots of a quadratic equation are the values of 'x' that make the equation true. Think of them as the points where the parabola (the graph of a quadratic equation) intersects the x-axis.

Now, why is this important? Well, quadratic equations pop up everywhere in the real world! They're used in physics to describe projectile motion, in engineering to design bridges and buildings, and even in finance to model investment growth. So, understanding how to solve them is a pretty valuable skill.

Why the Quadratic Formula?

Okay, so we know what a quadratic equation is. But why do we need a special formula to solve them? Can't we just, like, guess the answers? While guessing might work for very simple equations, it's definitely not a reliable method for anything slightly more complex. We need a systematic way to find those roots, and that's where the quadratic formula comes in.

The quadratic formula is a universal tool for solving quadratic equations. It works for any quadratic equation, no matter how messy the coefficients are. It's derived by completing the square on the general form of the quadratic equation (which is a whole other topic we could dive into!), but for now, let's just focus on how to use it. The quadratic formula states that for an equation in the form ax2+bx+c=0ax^2 + bx + c = 0, the roots are given by:

x=−b±b2−4ac2ax = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}

See? Not so scary, right? It might look a bit intimidating at first, but once you've used it a few times, it'll become second nature. The ± symbol simply means that there are two possible solutions: one where you add the square root part, and one where you subtract it. This makes sense because quadratic equations can have up to two distinct roots.

Applying the Quadratic Formula to Our Equation

Alright, let's get our hands dirty and actually use the quadratic formula to solve our equation: x2+16=10xx^2 + 16 = 10x.

Step 1: Rearrange the Equation

The first thing we need to do is rewrite the equation in the standard form ax2+bx+c=0ax^2 + bx + c = 0. To do this, we need to subtract 10x from both sides:

x2−10x+16=0x^2 - 10x + 16 = 0

Now we're talking! The equation is in the perfect form for applying the quadratic formula.

Step 2: Identify a, b, and c

Next, we need to identify the coefficients a, b, and c. Remember, 'a' is the coefficient of the x2x^2 term, 'b' is the coefficient of the x term, and 'c' is the constant term. In our equation:

  • a = 1 (because the coefficient of x2x^2 is 1)
  • b = -10 (notice the negative sign!)
  • c = 16

It's super important to get these values right, otherwise the quadratic formula won't give you the correct answer. Double-check your signs and make sure you're grabbing the coefficients from the correct terms.

Step 3: Plug the Values into the Quadratic Formula

Now for the fun part! We're going to plug our values of a, b, and c into the quadratic formula:

x=−(−10)±(−10)2−4(1)(16)2(1)x = \frac{-(-10) ± \sqrt{(-10)^2 - 4(1)(16)}}{2(1)}

Whoa, that looks like a mouthful! But don't panic. We'll take it one step at a time. Notice how we carefully substituted each value, including the negative signs. This is crucial to avoid errors.

Step 4: Simplify the Expression

Now comes the simplification stage. Let's break it down piece by piece:

  • −(−10)-(-10) becomes 10
  • (−10)2(-10)^2 becomes 100
  • 4(1)(16)4(1)(16) becomes 64
  • 2(1)2(1) becomes 2

So our equation now looks like this:

x=10±100−642x = \frac{10 ± \sqrt{100 - 64}}{2}

Much better, right? Let's keep simplifying:

x=10±362x = \frac{10 ± \sqrt{36}}{2}

Ah, a perfect square! We know that the square root of 36 is 6, so:

x=10±62x = \frac{10 ± 6}{2}

Step 5: Calculate the Two Roots

Remember that ± symbol? It means we have two possible solutions. Let's calculate them separately:

  • Root 1 (using the + sign): x=10+62=162=8x = \frac{10 + 6}{2} = \frac{16}{2} = 8

  • Root 2 (using the - sign): x=10−62=42=2x = \frac{10 - 6}{2} = \frac{4}{2} = 2

So, there you have it! The roots of the equation x2+16=10xx^2 + 16 = 10x are x = 8 and x = 2.

Expressing the Roots in Simplest Form

We've found our roots, but the question asks us to express them in the simplest form. In this case, our roots are already integers (whole numbers), so they are in their simplest form. If we had ended up with fractions or radicals, we would need to simplify them further.

For example, if we had a root like 42\frac{4}{2}, we would simplify it to 2. Or, if we had a root like 8\sqrt{8}, we would simplify it to 222\sqrt{2}.

The Discriminant: A Sneak Peek into the Roots

Before we wrap up, let's talk about a cool little part of the quadratic formula called the discriminant. The discriminant is the expression under the square root: b2−4acb^2 - 4ac. It gives us information about the nature of the roots without actually solving the equation.

  • If the discriminant is positive (b2−4ac>0b^2 - 4ac > 0), the equation has two distinct real roots (like our example).
  • If the discriminant is zero (b2−4ac=0b^2 - 4ac = 0), the equation has one real root (a repeated root).
  • If the discriminant is negative (b2−4ac<0b^2 - 4ac < 0), the equation has two complex roots (involving imaginary numbers).

In our case, the discriminant was 100−64=36100 - 64 = 36, which is positive, confirming that we have two distinct real roots.

Conclusion: Mastering the Quadratic Formula

And that's a wrap, guys! We've successfully solved a quadratic equation using the quadratic formula and expressed the roots in their simplest form. Remember, the key is to break it down step by step: rearrange the equation, identify a, b, and c, plug the values into the formula, simplify, and calculate the roots.

The quadratic formula might seem intimidating at first, but with practice, it'll become a powerful tool in your mathematical arsenal. So, keep practicing, and you'll be solving quadratic equations like a pro in no time! And remember, understanding the discriminant can give you valuable insights into the nature of the roots before you even start solving. Happy solving!