SOLUTION: Al-Khwarizmi solved all quadratic equations by reducing them to one of six standard forms, which were then easily solvable. He described the standard forms in terms of "squares," "

Question 1183944: Al-Khwarizmi solved all quadratic equations by reducing them to one of six standard forms, which were then easily solvable. He described the standard forms in terms of "squares," "roots," and "numbers." Here are al-Khwarizmi's six standard forms:
squares equal to roots (Example: ax2= bx
a x squared equals space b x)
squares equal to numbers (Example: ax2= c
a x squared equals space c)
roots equal to numbers (Example: bx=c
b x equals c)
squares and roots equal to numbers (Example: ax2+bx=c
a x squared plus b x equals c)
squares and numbers equal to roots (Example: ax2+c=bx
a x squared plus c equals b x)
roots and numbers equal tot squares (Example: ax2=bx+c
a x squared equals b x plus c)
Activity Instructions
• Which method would you use to solve each of the six forms? Why would you use that method?
• Write a quadratic equation that can be reduced to one of al-Khwarizmi's six forms.
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
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.

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