Question 1183944
Let's break down each of al-Khwarizmi's six forms and discuss appropriate solution methods, along with example equations.

**1. Squares equal to roots (ax² = bx):**

* **Method:** Divide both sides by 'x' (assuming x ≠ 0) to get ax = b, then solve for x: x = b/a.  We can also factor it as x(ax-b)=0, so x=0 or x=b/a.
* **Why:** This simplifies the quadratic to a linear equation, which is easy to solve.
* **Example:** 3x² = 12x  =>  3x = 12  =>  x = 4 (or x=0).

**2. Squares equal to numbers (ax² = c):**

* **Method:** Divide both sides by 'a' to get x² = c/a, then take the square root of both sides: x = ±√(c/a).
* **Why:** This isolates x², allowing us to directly find the value(s) of x using the inverse operation (square root).
* **Example:** 5x² = 20  =>  x² = 4  =>  x = ±2.

**3. Roots equal to numbers (bx = c):**

* **Method:** Divide both sides by 'b' to get x = c/b.
* **Why:** This is already a linear equation; one simple division gives the solution.
* **Example:** 7x = 21  =>  x = 3.

**4. Squares and roots equal to numbers (ax² + bx = c):**

* **Method:** This is the classic quadratic equation form. Use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a.  Completing the square also works.
* **Why:** The quadratic formula provides a general solution for this type of equation.
* **Example:** 2x² + 5x = 12.

**5. Squares and numbers equal to roots (ax² + c = bx):**

* **Method:** Rearrange the equation to the standard quadratic form (ax² - bx + c = 0) and then use the quadratic formula: x = (b ± √(b² - 4ac)) / 2a. Completing the square also works.
* **Why:** Similar to the previous case, the quadratic formula is a direct way to find the solution(s).
* **Example:** 3x² + 4 = 8x  =>  3x² - 8x + 4 = 0.

**6. Roots and numbers equal to squares (ax² = bx + c):**

* **Method:** Rearrange the equation to the standard quadratic form (ax² - bx - c = 0) and use the quadratic formula: x = (b ± √(b² + 4ac)) / 2a. Completing the square also works.
* **Why:** Again, the quadratic formula provides a general solution.
* **Example:** 2x² = 5x + 3  =>  2x² - 5x - 3 = 0.