Solving the Equation 𝑥^2−𝑥^3=2. A very nice olympiad question | How to solve (x²-x³ = 2) | Algebra | A-MATHS

Algebra can often present us with intriguing challenges that require a blend of creativity and technique to solve. One such interesting problem is the equation x² — x³ = 2. In this blog post, we’ll break down the steps to find the solution to this equation.

Step 1: Rearrange the Equation

First, let’s rewrite the equation in a more standard form. Starting with:
x² — x³ = 2

We can rearrange this to:
-x³ + x² — 2 = 0

Multiplying through by -1 for a cleaner look:
x³ — x² + 2 = 0

Step 2: Analyze for Real Solutions

To solve the equation x³ — x² + 2 = 0 , we can start by checking if there are any obvious real solutions by trial and error or using numerical methods.

Step 3: Use the Rational Root Theorem

The Rational Root Theorem states that any potential rational solution of a polynomial equation is a factor of the constant term divided by a factor of the leading coefficient. Here, the constant term is 2 and the leading coefficient is 1, so possible rational roots could be pm 1, pm 2 .

Let’s test these:

1.Testing x = 1:
1³ — 1² + 2 = 1–1 + 2 = 2
So, x = 1 is not a solution.

2. Testing x = -1 :
(-1)³ — (-1)² + 2 = -1–1 + 2 = 0
Hence, x = -1 is indeed a solution.

Step 4: Factoring the Polynomial

Since x = -1 is a root, we can factor x + 1 out of the polynomial x³ — x² + 2 To factor, we’ll perform polynomial division or use synthetic division:

Dividing x³ — x² + 2 by x + 1 ,

we get:
(x + 1)(x² — 2x + 2) = 0

Thus, the polynomial can be factored as:
x³ — x² + 2 = (x + 1)(x² — 2x + 2)

Step 5: Solving the Quadratic

Now, we need to solve x² — 2x + 2 = 0 . This quadratic equation can be solved using the quadratic formula x=(-b +- underroot(b²-4ac)/ 2a

Here, a = 1 ,b = -2 , and c = 2 :
\[ x = \frac{2 \pm \sqrt{(-2)² — 4 \cdot 1 \cdot 2}}{2 \cdot 1} = \frac{2 \pm \sqrt{4–8}}{2} = \frac{2 \pm \sqrt{-4}}{2} = \frac{2 \pm 2i}{2} = 1 \pm i \]

Step 6: Interpret the Solutions

Thus, the solutions to the equation x² — x³ = 2 are:
\[ x = -1, \quad x = 1 + i, \quad x = 1 — i \]

Conclusion

The equation x² — x³ = 2 has one real solution and two complex solutions:
- Real solution: x = -1
- Complex solutions: x = 1 + i and x = 1 — i

Understanding how to manipulate and factor polynomials, and how to use the quadratic formula, is essential in solving such equations. This problem is a great example of combining these algebraic techniques to find both real and complex roots. Happy solving!