(x^2-5x+7)^2-(x-2)(x-3)=1

3 min read Jun 17, 2024

Solving the Equation: (x^2 - 5x + 7)^2 - (x - 2)(x - 3) = 1

This article will guide you through the process of solving the equation (x² - 5x + 7)² - (x - 2)(x - 3) = 1. We'll break down each step to make it easy to understand.

Step 1: Expand the equation

First, we need to expand the equation to simplify it. Let's start by expanding the square term:

(x² - 5x + 7)² = (x² - 5x + 7)(x² - 5x + 7)

Using the distributive property (or FOIL method), we get:

(x² - 5x + 7)² = x⁴ - 10x³ + 39x² - 70x + 49

Now, expand the product of the two linear terms:

(x - 2)(x - 3) = x² - 5x + 6

Substitute these expanded expressions back into the original equation:

x⁴ - 10x³ + 39x² - 70x + 49 - (x² - 5x + 6) = 1

Step 2: Simplify the equation

Combining like terms, we get:

x⁴ - 10x³ + 38x² - 65x + 42 = 1

Subtract 1 from both sides:

x⁴ - 10x³ + 38x² - 65x + 41 = 0

Step 3: Factor the equation

Now we have a quartic equation. Factoring quartic equations can be challenging, and in this case, there isn't an obvious factorization. Therefore, we'll need to employ numerical methods or graphing tools to find the solutions.

Step 4: Using numerical methods or graphing tools

Numerical methods: Techniques like the Newton-Raphson method or the bisection method can be used to approximate the roots of the equation. These methods require an initial guess and iterate until a solution is found within a desired tolerance.
Graphing tools: Plotting the function y = x⁴ - 10x³ + 38x² - 65x + 41 will show you where the graph intersects the x-axis, which represents the roots of the equation.

Conclusion

Solving the equation (x² - 5x + 7)² - (x - 2)(x - 3) = 1 involves expanding, simplifying, and then employing numerical methods or graphing tools to find the solutions. The exact solutions are likely to be irrational numbers, requiring approximation techniques for practical use.