(x-a-7)(x+a-2)/sqrt(10x-x^2-a^2)=0

5 min read Jun 17, 2024

Solving the Equation: (x-a-7)(x+a-2)/sqrt(10x-x^2-a^2)=0

This equation presents a unique challenge due to the presence of a square root in the denominator. Let's break down the steps to solve it:

Understanding the Equation

The equation is a rational expression where the numerator is a product of two linear factors: (x-a-7) and (x+a-2). The denominator is the square root of a quadratic expression: sqrt(10x-x^2-a^2).

Key Points:

Rational Expression: A rational expression is a fraction where both the numerator and denominator are polynomials.
Square Root: The denominator involves a square root, meaning the expression is undefined when the radicand (the expression inside the square root) is negative.
Zero Product Property: A product equals zero if and only if at least one of the factors is zero.

Solving for x

Set the Numerator to Zero:

To make the entire fraction zero, the numerator must be zero. Therefore, we set:

(x-a-7)(x+a-2) = 0

Applying the zero product property, we get two possible solutions:
- x - a - 7 = 0 => x = a + 7
- x + a - 2 = 0 => x = 2 - a
Check the Denominator:

We need to ensure that the denominator is not zero, as division by zero is undefined. This means:

sqrt(10x-x^2-a^2) ≠ 0

Squaring both sides, we get:

10x - x^2 - a^2 ≠ 0

Rearranging, we get:

x^2 - 10x + a^2 ≠ 0

This quadratic inequality needs to be solved to find the values of x that make the denominator non-zero. However, we can't find a general solution for this without knowing the specific value of 'a'.
Combining the Solutions:

We have two potential solutions for 'x' from step 1 (x = a + 7 and x = 2 - a). We need to check if these solutions make the denominator non-zero, taking into account the quadratic inequality from step 2. If either solution makes the denominator zero, it is an extraneous solution and must be discarded.

Example: Finding Solutions for a Specific Value of 'a'

Let's consider a specific example where a = 3:

x = a + 7 = 3 + 7 = 10
x = 2 - a = 2 - 3 = -1

Now we need to check if these values satisfy the inequality:

For x = 10: 10^2 - 10(10) + 3^2 = 9 ≠ 0. This solution is valid.
For x = -1: (-1)^2 - 10(-1) + 3^2 = 22 ≠ 0. This solution is also valid.

Therefore, for a = 3, the solutions to the equation are x = 10 and x = -1.

Conclusion

Solving the equation (x-a-7)(x+a-2)/sqrt(10x-x^2-a^2)=0 requires careful consideration of the numerator, denominator, and the zero product property. The final solution will depend on the specific value of 'a'. Remember to always check for extraneous solutions by ensuring the denominator does not become zero.