Final answer:
To create an equation with extraneous solutions of the form x+a=v-----bx+c, you can choose values for a, b, and c that make the equation true for a certain value of x and then square both sides of the equation to create extraneous solutions.
Step-by-step explanation:
To create an equation of the form x+a=v-----bx+c with extraneous solutions, we need to choose values for a, b, and c that will make the equation true for a particular value of x, but also yield extraneous solutions when solving the equation. To demonstrate this, let's choose a = 3, b = 13, and c = -10. Our equation becomes x + 3 = v ----- 13x - 10.
To solve this equation, we can substitute x = 7 and solve for v. When we do this, we get 7 + 3 = v. Therefore, v = 10. So one solution of the equation is x = 7 and v = 10.
Now, let's square both sides of the equation x + 3 = v ----- 13x - 10 to create an equation that will yield extraneous solutions. Squaring both sides gives us (x + 3)² = (v ----- 13x - 10)². When we expand this equation, we get x² + 6x + 9 = v² - 26vx + 169x² - 20v + 100. Simplifying further, we get x² + (6 + 26v - 169)x + (9 + 20v - 100) = 0.