## An 1851 Quadratic Factoring Method

Posted by mark schwartz on April 7, 2016

I’ve never seen anything like this. In *A Theoretical and Practical Treatise on Algebra *(1851), Mr. Robinson demonstrates a factoring method that may have been common at that time, but in the set of old texts I have, no other author has presented it. I can see why it may have gone out of use, since there are more steps than current procedures. But nonetheless, it’s worth getting a sense of how some students were taught to factor in 1851.

This procedure applies to quadratics of the form ax^{2} + bx + c, so be sure to remove all common factors first and if a ≠1, divide all terms by the coefficient of x^{2}.

Take the quadratic x^{2 }+ 7x + 12.

Step 1. The sum of two numbers gives the coefficient of the “x” term, so, a + b = 7.

Step 2. The product of two numbers gives the constant term, so ab = +12.

Step 3. Square both sides of the equation a + b = 7, giving a^{2} + 2ab + b^{2} = 49.

Step 4. Multiply “ab = +12” by 4, giving 4ab = 48.

Step 5. Subtract 4ab = 48 from a^{2} + 2ab + b^{2} = 49, giving a^{2} ̶ 2ab + b^{2} = 1.

Step 6. a^{2} ̶ 2ab + b^{2} = 1; therefore (a ̶ b)^{2} = 1.

Step 7. Take the square root of both sides of (a ̶ b)^{2} = 1, giving a ̶ b = ± 1.

Step 8. Add a + b = 7 and a ̶ b = + 1. The result is a = 4 and b = 3.

Step 9. Subtract a ̶ b = ̶ 1 from a + b = 7. The result is a = 3 and b = 4.

Mr. Robinson simply states that given these values for “a” and “b”, these are also the values for x. He doesn’t make it clear why this is true … but it is. Basically, his method has captured the quadratic formula but in a somewhat round-about way.

You’re done. See? It is interesting but as I say, there are too many steps compared to the number of steps in today’s procedures.

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