Solving $x^2 - 20x + 100 = 0$: A Step-by-Step Guide

Alright, guys, let's dive into solving this quadratic equation! We're given $x^2 - 20x + 100 = 0$, and our mission is to find the roots, or the values of x that make this equation true. Buckle up, because we're about to break it down.

Factoring the Quadratic Equation

First off, we want to factor the quadratic equation. Factoring is like reverse-engineering the equation to find two expressions that, when multiplied together, give us the original equation. In our case, we're looking for two numbers that add up to -20 (the coefficient of the x term) and multiply to 100 (the constant term).

So, we start with:

$x^2 - 20x + 100 = 0$

We're trying to get it into this form: $(x + M)(x + N) = 0$, where M and N are the numbers we're hunting for. We need M + N to equal -20, and M * N to equal 100. Think of two numbers that fit the bill. How about -10 and -10? Let's check:

• M + N = -10 + (-10) = -20 (Perfect!)
• M * N = (-10) * (-10) = 100 (Nailed it!)

So, we can rewrite our equation as:

$(x - 10)(x - 10) = 0$

This is the factored form of our quadratic equation. Cool, right?

Finding the Roots

Now that we have the factored form, finding the roots is a piece of cake. We know that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero and solve for x:

$x - 10 = 0$

Add 10 to both sides:

$x = 10$

Since both factors are the same $(x - 10)$, we only get one unique root. That means our equation has a repeated root. So, the solution set is:

$x = {10}$

Therefore, the root of the equation $x^2 - 20x + 100 = 0$ is 10. It's the only value of x that satisfies the equation.

Why This Works: A Deeper Dive

You might be wondering why this factoring thing works. Well, it all comes down to the distributive property. When we expand $(x - 10)(x - 10)$, we get:

$(x - 10)(x - 10) = x(x - 10) - 10(x - 10) = x^2 - 10x - 10x + 100 = x^2 - 20x + 100$

See? We get back our original equation. Factoring is just undoing this process.

Also, recognizing perfect square trinomials can speed things up. A perfect square trinomial is in the form $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$. These can be factored into $(a + b)^2$ or $(a - b)^2$, respectively. In our case, $x^2 - 20x + 100$ is a perfect square trinomial because it fits the form $a^2 - 2ab + b^2$, where $a = x$ and $b = 10$.

Alternative Method: The Quadratic Formula

If factoring isn't your cup of tea, or if you're dealing with a quadratic equation that's hard to factor, you can always use the quadratic formula. The quadratic formula is a universal tool for finding the roots of any quadratic equation in the form $ax^2 + bx + c = 0$. The formula is:

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

In our equation, $x^2 - 20x + 100 = 0$, we have $a = 1$, $b = -20$, and $c = 100$. Plugging these values into the quadratic formula, we get:

$x = \frac{-(-20) ± \sqrt{(-20)^2 - 4(1)(100)}}{2(1)}$

Simplify:

$x = \frac{20 ± \sqrt{400 - 400}}{2}$

$x = \frac{20 ± \sqrt{0}}{2}$

$x = \frac{20 ± 0}{2}$

$x = \frac{20}{2}$

$x = 10$

Again, we find that the only root is $x = 10$. The quadratic formula works every time, even when factoring seems tricky!

Common Mistakes to Avoid

• Sign Errors: Pay close attention to the signs of the coefficients when factoring or using the quadratic formula. A simple sign error can throw off your entire solution.
• Incorrect Factoring: Double-check your factoring by expanding the factors to make sure you get back the original equation.
• Forgetting the ± in the Quadratic Formula: Remember that the quadratic formula gives you two possible solutions because of the ± sign. In some cases, like ours, the two solutions might be the same, but it's important to consider both possibilities.
• Arithmetic Errors: Be careful with your arithmetic, especially when dealing with square roots and fractions. A small calculation error can lead to an incorrect answer.

Real-World Applications

Quadratic equations aren't just abstract math problems; they show up in all sorts of real-world situations. For example, they're used in physics to model the trajectory of a projectile, in engineering to design bridges and buildings, and in finance to calculate compound interest. Understanding how to solve quadratic equations is a valuable skill that can help you in many different fields.

Practice Problems

Want to sharpen your skills? Try solving these quadratic equations:

1. $x^2 - 6x + 9 = 0$
2. $x^2 + 8x + 16 = 0$
3. $4x^2 - 12x + 9 = 0$

See if you can factor them or use the quadratic formula to find the roots. Good luck, you got this!

Conclusion

So, there you have it! We've successfully solved the quadratic equation $x^2 - 20x + 100 = 0$ by factoring and using the quadratic formula. Remember, practice makes perfect, so keep working at it, and you'll become a quadratic equation-solving pro in no time!

Solving quadratic equations might seem daunting at first, but with a little practice, you'll be able to tackle them with confidence. Whether you prefer factoring, using the quadratic formula, or a combination of both, the key is to understand the underlying concepts and to be careful with your calculations. And who knows, maybe one day you'll even use your quadratic equation-solving skills to build a bridge or launch a rocket! Keep learning and keep exploring, and you'll be amazed at what you can achieve.