(x^2+x-1)(x^2+x+2)=40

3 min read Jun 17, 2024

Solving the Equation (x^2 + x - 1)(x^2 + x + 2) = 40

This problem involves solving a quartic equation, which can be a bit tricky. Let's break down the steps:

1. Expand the Equation

First, expand the left side of the equation by multiplying the two quadratic expressions:

(x^2 + x - 1)(x^2 + x + 2) = 40 
x^4 + x^3 + 2x^2 + x^3 + x^2 + 2x - x^2 - x - 2 = 40
x^4 + 2x^3 + 2x^2 + x - 2 = 40

2. Simplify the Equation

Now, move all terms to one side to get a standard quartic equation:

x^4 + 2x^3 + 2x^2 + x - 42 = 0

3. Finding Solutions

Solving a quartic equation directly can be quite challenging. There are a few ways to approach this:

Factoring: Try to factor the equation. In this case, it's unlikely to be factorable easily.
Rational Root Theorem: This theorem can help find potential rational roots. However, applying it to this equation doesn't yield any simple rational roots.
Numerical Methods: Using numerical methods like the Newton-Raphson method or graphing calculators can give approximate solutions.
Substitution: We can try a substitution to simplify the equation. Let's substitute y = x^2 + x. This gives us:
```
y(y + 3) - 42 = 0
y^2 + 3y - 42 = 0
```
This is a quadratic equation, which we can solve using the quadratic formula:
```
y = (-3 ± √(3^2 - 4 * 1 * -42)) / (2 * 1)
y = (-3 ± √177) / 2
```
Now, we need to substitute back x^2 + x for y and solve for x:
```
x^2 + x = (-3 ± √177) / 2
```
This leads to two quadratic equations which can be solved using the quadratic formula again.

4. Solutions

The equation (x^2 + x - 1)(x^2 + x + 2) = 40 has four solutions, which are the roots of the resulting quadratic equations after the substitution. You can use a calculator or a computer algebra system to find the approximate values of these solutions.

Note: The solutions are not simple integers or fractions, they are likely irrational numbers.