Algebra: How do YOU solve for 10x^2+x-21=0?

Do a whole bunch of other stuff with graphing designed to make your eyes glaze over.

Someone I know IRL recently wrote, “I didn’t use algebra at all today.” That was probably not true, but no matter.

The Daughter is studying algebra right now in 8th grade and it’s a real PITA. Because she was so good in 7th grade math, she skipped over 8th grade math, what that was, and is now taking the math for 9th grade. This is a problem because she doesn’t know, and I surely don’t remember, what she’s missing.

I should note that when I was in 9th grade, I was very good in algebra. I remember helping a fellow student, Sid, at the chalkboard, when Miss McNulty couldn’t get him to understand.

I got a 97 in the Regents final. (Yes, I remember this; no I didn’t look it up. I got 86 in geometry and 98 is trigonometry.) But that was a HALF CENTURY AGO. THAT will make you feel old.

I have been depending on something called Tiger Algebra to help her muddle through.

For the problem 10x^2+x-21=0, where the ^ over the 6 key represents “power of,” so ten X squared in this case.

The factoring is the tough part to explain.

Factoring 10x^2+x-21
The first term is 10x^2 – its coefficient is 10
The middle term is +x its coefficient is 1
The last term, “the constant”, is -21

Step-1 : Multiply the coefficient of the first term by the constant 10 • -21 = -210

Step-2 : Find two factors of -210 whose sum equals the coefficient of the middle term, which is 1 .

-210 + 1 = -209
-105 + 2 = -103
-70 + 3 = -67
-42 + 5 = -37
-35 + 6 = -29
-30 + 7 = -23
-21 + 10 = -11
-15 + 14 = -1
-14 + 15 = 1 That’s it

By “it,” we’re talking the very beginning of “it.”

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -14 and 15
10x^2 – 14x + 15x – 21

Step-4 : Add up the first 2 terms, pulling out like factors :
2x • (5x-7)
Add up the last 2 terms, pulling out common factors :
3 • (5x-7)

Step-5 : Add up the four terms of step 4 :
(2x+3) • (5x-7)
Which is the desired factorization

At which point you take 2x+3=0 and 5x-7=0, and get 1.5 and 1.4 respectively, then do a whole bunch of other stuff with graphing designed to make your eyes glaze over.

We usually work on this in the morning, after the Daughter has felt despair the night before, which means doing it in lieu of me blogging in the morning, which is my best time for writing.

And she SORT OF understands parts of this. Hey, if you have an easier way to find the factors, please let me know. My blog will thank you, publicly if you want.