Question

Solve the radical equation. x-5= √ (-2x+18). What is the true solution to the radical equation? 1, 7, There are no true solutions, Both 1 and 7 are true solutions

a) x=1
b) x=7
c) There are no true solutions
d) Both 1 and 7 are true solutions

Answer 1

To solve the radical equation x - 5 = √(-2x + 18), we square both sides, simplify, and factor to find potential solutions x = 1 and x = 7. Upon checking these values, only x = 7 satisfies the original equation, making it the true solution.

Step-by-step explanation:

The student's question asks us to solve the radical equation x - 5 = √(-2x + 18). To find the true solution of this equation, we will first square both sides to eliminate the square root:

1. [(x - 5)]² = [√(-2x + 18)]²
2. x² - 10x + 25 = -2x + 18
3. x² - 8x + 7 = 0

This is a quadratic equation in standard form, which can be factored to find the solutions for x:

1. (x - 1)(x - 7) = 0
2. Therefore, x = 1 or x = 7.

However, we must check these potential solutions to make sure they satisfy the original equation:

1. For x = 1: 1 - 5 ≠ √(-2(1) + 18), which is -4 ≠ 4, not true.
2. For x = 7: 7 - 5 = √(-2(7) + 18), which is 2 = 2, true.

Thus, the true solution to the radical equation is x = 7, which means the correct answer is option b).