Equation shortcut

I was doing some math recently when I realized something was odd about a particular equation.

ax^2 + bx = 0
a times x squared plus b times x equals zero. This equation would normally be solved by completing the square and yada-yada-yada. However, I couldn’t help but notice the strange answer I got from this.

x^2 + 36x = 0
x = 0 or x = -36

This got me thinking, and after playing around on my calculator for a bit, I came to the conclusion of this handy shortcut.

Given ax^2 + bx = 0, x = 0 or x = -b/a

Try it out for yourself, no squaring and other crap needed.

5x^2 – 35x = 0

From this we know, x = 0 or x = 7

If we had to do it by completing the square…

1. 5(x^2 – 7 + 49/4) = 254/4
2. (x – 7/2)^2 = + or – sqrt(49/4)
3. x – 7/2 = 7/2 or x – 7/2 = -7/2
4. x = 7 or x = 0

Or the quadratic formula

1. 35 + or – sqrt((-35)^2 – 4(5)(0)) / 2(5)
2. 35 + or – sqrt(1225) / 10
3. 35 + or – 35 / 10
4. 7 + or – 7 / 2
5. x = (7 + 7)/2 = 7 or x = (7 – 7)/2 = 0

This shortcut answer does seem to have a relationship with the quadratic formula and vertex formula of a parabola, but I assume that since we’re accounting for 2 values of change and not 3, that that’s why you only have to divide by a and not by 2a. Not sure really though. I looked up and down for this shortcut on the internet and couldn’t find anything, so I decided to write about it.

 
 
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