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Question 1181912: 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-khwarizmis six standard forms.
1. Squares equal to roots(example:ax^2=bx)
2. Squares equal to numbers(example:ax^2=c)
3.roots equal to numbers(example:bx=c)
4.squares and roots equal to numbers (example:ax^2+bx=c)
5. Squares and numbers equal to roots (example: ax^2+c=bx)
6.roots and numbers equal to tot squares(example:ax^2=bx+c)
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-khwarizmis six forms.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's a breakdown of how to solve each of Al-Khwarizmi's six forms, along with explanations:
**1. Squares equal to roots (ax² = bx)**
* **Method:** Divide both sides by *x* (assuming *x* is not zero). This simplifies the equation to *ax = b*, which can be solved directly for *x*.
* **Why:** Dividing by *x* reduces the quadratic to a linear equation, which is much easier to solve. We must consider the case where *x* = 0 separately, as it is a solution to the original equation.
**2. Squares equal to numbers (ax² = c)**
* **Method:** Divide both sides by *a*, then take the square root of both sides. Remember to consider both the positive and negative square roots. This gives *x* = ±√(c/a).
* **Why:** Taking the square root isolates *x*. The ± is crucial because both positive and negative values, when squared, can equal *c/a*.
**3. Roots equal to numbers (bx = c)**
* **Method:** Divide both sides by *b*. This gives *x = c/b*.
* **Why:** This is already a linear equation; dividing by *b* directly solves for *x*.
**4. Squares and roots equal to numbers (ax² + bx = c)**
* **Method:** Complete the square. Divide the entire equation by *a* to get *x² + (b/a)x = c/a*. Take half of the coefficient of *x* (which is *b/2a*), square it (*b²/4a²*), and add it to both sides. This creates a perfect square on the left side: *x² + (b/a)x + b²/4a² = c/a + b²/4a²*. Rewrite the left side as *(x + b/2a)² = c/a + b²/4a²*. Then take the square root of both sides, remembering the ±, and solve for *x*.
* **Why:** Completing the square transforms the quadratic into a form where taking the square root isolates *x*.
**5. Squares and numbers equal to roots (ax² + c = bx)**
* **Method:** Rearrange the equation to get it into the standard form for completing the square (ax² - bx + c = 0). Then, follow the same "completing the square" steps as in form 4.
* **Why:** Same as form 4, completing the square makes the equation solvable by taking the square root.
**6. Roots and numbers equal to squares (ax² = bx + c)**
* **Method:** Rewrite as ax² - bx - c = 0. Then, again, complete the square as in forms 4 and 5.
* **Why:** Consistent with the previous forms, completing the square is the key to isolating x.
**Example Quadratic Equation:**
The equation 3x² + 5 = 7x can be reduced to Al-Khwarizmi's form 5 (Squares and numbers equal to roots). Subtracting 7x from both sides gives 3x² - 7x + 5 = 0.
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