It’s one of those math problems that looks a bit daunting at first glance – a cubic equation, specifically x³ + 4x² = 5. You might be staring at it, wondering where to even begin. But like many things in mathematics, breaking it down makes it much more approachable.
Let's reframe the equation slightly. We want to find the values of 'x' that make this statement true. The first step is usually to get everything on one side, setting the equation to zero. So, we can rewrite x³ + 4x² = 5 as x³ + 4x² - 5 = 0.
Now, for cubic equations, there isn't always a straightforward formula like the quadratic formula for second-degree equations. However, we can often find integer roots by testing simple values. We're looking for a number that, when plugged into the equation, makes it equal zero. Let's try some small integers.
What if x = 1? Plugging that in, we get (1)³ + 4(1)² - 5 = 1 + 4 - 5 = 0. Bingo! We found one solution: x = 1.
Finding one root is a huge step because it means we can factor the cubic polynomial. Since x = 1 is a root, (x - 1) must be a factor of x³ + 4x² - 5. We can use polynomial division or synthetic division to find the other factor.
Let's use synthetic division with our root, 1, and the coefficients of our polynomial (1 for x³, 4 for x², 0 for x, and -5 for the constant term):
1 | 1 4 0 -5
| 1 5 5
----------------
1 5 5 0
The result (1, 5, 5) gives us the coefficients of the remaining quadratic factor: x² + 5x + 5.
So, our original equation x³ + 4x² - 5 = 0 can be factored into (x - 1)(x² + 5x + 5) = 0.
Now we have two parts to solve:
- x - 1 = 0: This gives us our first solution, x = 1.
- x² + 5x + 5 = 0: This is a quadratic equation. We can solve this using the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a.
In this case, a = 1, b = 5, and c = 5.
Plugging these values into the formula: x = [-5 ± √(5² - 4 * 1 * 5)] / (2 * 1) x = [-5 ± √(25 - 20)] / 2 x = [-5 ± √5] / 2
This gives us two more solutions:
x = (-5 + √5) / 2 x = (-5 - √5) / 2
So, the cubic equation x³ + 4x² = 5 has three solutions: x = 1, x = (-5 + √5) / 2, and x = (-5 - √5) / 2. It’s a good reminder that even seemingly complex problems can be unraveled with a systematic approach and a bit of patience.
