(x+4)(x+5)(x+7)(x+8)-4=0

2 min read Jun 16, 2024

Solving the Equation (x+4)(x+5)(x+7)(x+8) - 4 = 0

This equation looks complex, but we can solve it using a clever approach! Here's how:

Recognizing a Pattern

Let's focus on the first part of the equation: (x+4)(x+5)(x+7)(x+8). Notice that the terms within the parentheses are consecutive numbers. This suggests a pattern we can exploit.

A Useful Substitution

Let's make a substitution to simplify the expression. Let:

y = x + 6

Now we can rewrite the equation as:

(y-2)(y-1)(y+1)(y+2) - 4 = 0

Expanding and Simplifying

Expanding the first four terms, we get:

(y² - 4)(y² - 1) - 4 = 0

Expanding further:

y⁴ - 5y² + 4 - 4 = 0

This simplifies to:

y⁴ - 5y² = 0

Solving the Quadratic

We can factor out a y²:

y²(y² - 5) = 0

This gives us two possible solutions:

y² = 0 => y = 0
y² - 5 = 0 => y² = 5 => y = ±√5

Finding the Values of x

Remember that we substituted y = x + 6. Let's substitute back to find the values of x:

y = 0: 0 = x + 6 => x = -6
y = √5: √5 = x + 6 => x = √5 - 6
y = -√5: -√5 = x + 6 => x = -√5 - 6

The Solutions

Therefore, the solutions to the equation (x+4)(x+5)(x+7)(x+8) - 4 = 0 are:

x = -6
x = √5 - 6
x = -√5 - 6