(x+6)(x+7)(x+9)(x+10)=10

4 min read Jun 17, 2024

Solving the Equation (x+6)(x+7)(x+9)(x+10) = 10

This equation looks a bit intimidating at first glance, but we can solve it by using a few clever tricks and techniques. Here's how we can approach it:

1. Expanding and Simplifying

Let's start by expanding the left side of the equation. It might seem like a lot of work, but it's easier than it looks. We can use the distributive property and focus on expanding two pairs of parentheses at a time:

Step 1: Expand (x+6)(x+7) = x² + 13x + 42
Step 2: Expand (x+9)(x+10) = x² + 19x + 90
Step 3: Now, we have (x² + 13x + 42)(x² + 19x + 90) = 10. Expanding this further, we get a fourth-degree polynomial:

x⁴ + 32x³ + 361x² + 1650x + 3780 = 10
Step 4: Subtract 10 from both sides to set the equation to zero:

x⁴ + 32x³ + 361x² + 1650x + 3770 = 0

2. Finding Rational Roots (If Any)

This equation might have rational roots. We can use the Rational Root Theorem to find potential rational solutions. The theorem states that if a polynomial equation has a rational root p/q (where p and q are integers), then p must be a factor of the constant term (3770) and q must be a factor of the leading coefficient (1).

Factors of 3770: ±1, ±2, ±5, ±7, ±10, ±17, ±19, ±29, ±34, ±85, ±95, ±133, ±161, ±190, ±221, ±323, ±380, ±665, ±761, ±1145, ±1885, ±3770
Factors of 1: ±1

This gives us a lot of potential rational roots! We can try these values one by one by substituting them into the equation. Unfortunately, it turns out that none of these rational numbers are roots of the equation.

3. Using Numerical Methods

Since we haven't found any rational roots, we can use numerical methods like the Newton-Raphson method or bisection method to approximate the solutions. These methods involve iterative calculations to find increasingly accurate approximations of the roots.

Note: Solving this equation using these methods requires a calculator or computer software.

4. Graphing the Equation

Another approach is to graph the function y = x⁴ + 32x³ + 361x² + 1650x + 3770. The x-intercepts of the graph represent the real solutions (or roots) of the equation. A graphing calculator or software can help visualize the roots.

Conclusion

While finding exact solutions for the equation (x+6)(x+7)(x+9)(x+10) = 10 can be challenging, we can use various techniques to understand the equation and find its solutions. We can try finding rational roots, use numerical methods to approximate solutions, or graph the function to visualize the roots.