It's funny how a string of numbers and letters can sometimes feel like a puzzle, isn't it? Take this one: x(5x + 21) = 20. On the surface, it might look a bit daunting, especially if algebra wasn't your favorite subject back in school. But honestly, when you break it down, it's more like a friendly conversation with numbers.
Let's start by tidying things up. That initial equation, x(5x + 21) = 20, is a bit like a guest who hasn't quite settled in. We need to get it into a more standard form so we can understand it better. So, we distribute that 'x' on the left side: x times 5x gives us 5x², and x times 21 gives us 21x. Now we have 5x² + 21x = 20. To make it a proper quadratic equation, we want everything on one side, with zero on the other. So, we subtract 20 from both sides, and voilà: 5x² + 21x - 20 = 0.
This is where things get interesting. We've transformed our initial expression into a standard quadratic equation, often written as ax² + bx + c = 0. In our case, 'a' is 5, 'b' is 21, and 'c' is -20. Now, we have a couple of ways to figure out what 'x' can be.
One common approach is factoring. It's like finding two numbers that, when multiplied, give you a*c (which is 5 * -20 = -100) and, when added, give you 'b' (which is 21). Finding these two numbers can sometimes be a bit of a hunt, but with a little patience, we can see that 25 and -4 fit the bill. 25 * -4 = -100, and 25 + (-4) = 21. So, we can rewrite the middle term, 21x, as 25x - 4x. Our equation now looks like 5x² + 25x - 4x - 20 = 0. From here, we can group terms: (5x² + 25x) - (4x + 20) = 0. Factoring out common terms from each group gives us 5x(x + 5) - 4(x + 5) = 0. See that (x + 5) in both parts? That's a good sign! We can then factor out (x + 5) to get (5x - 4)(x + 5) = 0. For this product to be zero, either (5x - 4) must be zero or (x + 5) must be zero. Solving these simple linear equations, we find our solutions: x = 4/5 and x = -5.
Another way to tackle this, especially if factoring feels tricky, is using the quadratic formula. This formula is a lifesaver for any quadratic equation. It tells us that x = [-b ± √(b² - 4ac)] / 2a. Plugging in our values for a, b, and c (5, 21, and -20 respectively), we calculate the discriminant (b² - 4ac) first: 21² - 4 * 5 * (-20) = 441 + 400 = 841. The square root of 841 is 29. Now, we substitute this back into the formula: x = [-21 ± 29] / (2 * 5). This gives us two possibilities: x = (-21 + 29) / 10 = 8 / 10 = 4/5, and x = (-21 - 29) / 10 = -50 / 10 = -5. Exactly the same answers we got from factoring!
It's also neat to know that we can determine the nature of the roots without even solving for them. That discriminant, b² - 4ac, is a little indicator. Since our discriminant (841) is positive, we know there are two distinct real roots. If it were zero, there would be exactly one real root (or two equal roots), and if it were negative, there would be no real roots (but two complex ones).
So, from a seemingly simple expression, we've navigated through algebraic manipulation, factoring, and the power of the quadratic formula, all to find the values of 'x' that make the original equation true. It's a journey that shows how a bit of structured thinking can unlock the secrets hidden within mathematical expressions.
