777 Joined: Oct 7, 2015
• Posts: 724
Thanks for this post from: November 25th, 2020 at 1:58:58 PM permalink

I have this equation in the form of ax^2 + bx + c =0, where a = 1, b = -131, and c = 3870, and attempted to solve it using the new method, but wasn't successful. Is this equation quadratic?
ChesterDog Joined: Jul 26, 2010
• Posts: 997
November 25th, 2020 at 2:43:31 PM permalink
Quote: 777

I have this equation in the form of ax^2 + bx + c =0, where a = 1, b = -131, and c = 3870, and attempted to solve it using the new method, but wasn't successful. Is this equation quadratic?

I like that new method, and it worked for me on your equation.

The sum of the answers should be 131, which means they are in the form 131/2 - u and 131/2 + u. That is x = 65.5 - u and x = 65.5 + u.

The product of answers equals 3870:
(65.5-u)(65.5+u) = 65.52 - u2 = 3870

Therefore u2 = 4290.25 - 3870 = 420.25.

So, u = (+ or -)sqrt(420.25) = (+ or -)20.5.

Then x - u = 45, and x + u = 86.

ThatDonGuy Joined: Jun 22, 2011
• Posts: 5165
November 25th, 2020 at 2:46:50 PM permalink
Quote: 777

I have this equation in the form of ax^2 + bx + c =0, where a = 1, b = -131, and c = 3870, and attempted to solve it using the new method, but wasn't successful. Is this equation quadratic?

Yes, it is, and there's nothing really new about the method.

x^2 - 131 x + 3870 = 0 has the sum of its roots 131 and the product of its roots 3870.
The roots are 131/2 + z and 131/2 - z.
The product is (131/2)^2 - z^2= 3870
This is (131/2)^2 - 3870 = z^2
z^2 = (17,161 - 15,480) / 4 = 1681 / 4
z = 41 / 2

A similar way of doing this is called "completing the square": this stems from (x - a)^2 = x^2 - 2a x + a^2.
Subtract 3870 from both sides: x^2 - 131 x = -3870
Add (-131/2)^2 to both sides: x^2 - 131 x + (-131/2)^2 = (-131/2)^2 - 3870
The left side is (x - 131/2)^2; the right side is 17,161 / 4 - 15,480 / 4 = 1681 / 4 = (41 / 2)^2
Take the square root of both sides, keeping in mind that the square root of X^2 is both X and -X:
+/- (x - 131/2) = 41 / 2
x = 131 / 2 +/- 41 / 2 = 86, 45
(x - 131/2) = +/- sqrt(2041) / 2
x = 131/2 +/- sqrt(2041) / 2
chevy Joined: Apr 15, 2011
• Posts: 91
November 25th, 2020 at 3:05:47 PM permalink

I rather like the "new" method and feels it offers some insight in developing the original algebra skills of factoring quadratics (with nice answers) by guessing at roots and seeing which work. For actual determination of the roots, it is is literally just doing the two parts of quadratic equation separately.....where a=1

average = -b/(2a) = -b/2
and solving for
u= (+ or -) sqrt(b^2 - 4ac) / (2a)
= (+ or -) sqrt(b^2 - 4c) /2
= (+ or -) sqrt[ (b/2)^2 - c)]

roots are (average + u) and (average - u)

I think the manipulation of the steps solving for "u" is no easier for irrational or complex roots than plugging into the quadratic formula.

average = -b/2 = -(-131)/2 = 131/2

(131/2 + u)*(131/2 - u) = 3870

(131^2)/4 - u^2 = 3870

u^2 = (131^2)/4 - 3870 = 420.25

u=20.5

roots are
131/2+20.5 = 65.5+20.5 = 86
131/2-20.5 = 65.5-20.5 = 45