Math - Factoring Quadratics - Challenge

In this lecture we looked a simple way of factoring our equations, without having having to learn the rather cumbersome quadratic formula. This method helps us to convert our equation from standard form into intercept form and also tells up how many roots our equation will have.

For your challenge, convert the following equation into intercept, or factored, form;

y = -1/3x^2 + 10/3x - 16/3

Post your answer below and remember to use spoiler tags.

1 Like

y = -1/3x² + 10/3x - 16/3

-3y = x² - 10x + 16
10/2 = 5
so (5 + u)(5 - u) = 16
25 - u² = 16
u² = 25 - 16
u = √9
u = 3

5+3 = 8 and 5-3 = 2
so y = -1/3 (x - 2)(x-8)
we can see that (2,0) and (8,0) are our roots like before.

1 Like

y = -1/3 (x-2)(x-8) Roots: (2,0) and (8,0)

1 Like

-3y = x^2 - 10x + 16

(5+u)(5-u) = 16

25 - u^2 = 16

-u^2 = -9

u = 3

(5-3)(5+3) = 2 x 8 = 16

y = -1/3(x-2)(x-8)

1 Like

Yeah!! :slight_smile:

y = -1/3x^2 + 10/3x - 16/3
Convert to intercept form.
-3y = x^2 - 10x + 16

(-5-u)(-5+u)=16
25-u^2=16
u^2=3

-3y = (x-8)(x-2)

y = -1/3(x-8)(x-2)

1 Like

y=−(1/3) x^2+(10/3)x −(16/3) −Convert to intercept form
−3y=x^2−10x+16
(5+u)(5−u)=16
25−u^2=16
−u^2=−9
u=3
(5+3)∗(5−3)=16
(8)∗(2)=16
y=−(1/3)(x−8)(x−2)

1 Like

My results are:

y = -1/3x² + 10/3 - 16/3
-3y = x² - 10 + 16

(5 - u)(5 + u) = 16
25 - u² = 16
u² = 25 - 16
u = √9
u = 3

(5 - 3)(5 + 3) = 16
(2)(8) = 16

(2,0) (8,0)

I was able to follow this along using your method. But what I am not getting really is how we get from

(5 - u)(5 + u) to 25 - u²

and from

25 - u² = 16 to u² = 25 - 16!

It’s probably fairly simple but I am somehow missing it. If you have a minute could you explain that please?

Hi @TheSnooze,
We can get from (5 + u)(5 - u) to 25 - u^2 by expanding the parentheses.
For this, you can use the FOIL method.
Then it’s just a case of rearranging the equation.

Here are the full steps,

    (5 + u)(5 - u) = 16         <- expand parentheses using FOIL
25 - 5u + 5u - u^2 = 16         <- -5u + 5u cancel out
         -25 + u^2 = -16        <- multiply both sides by -1
               u^2 = -16 + 25   <- add 25 from both sides
                 u = √9         <- take the square root of both sides
                 u = 3          <- simplify

Once you get used to manipulating equations, you’ll find that there are a few shortcuts that you can take to speed things up - Such as expanding binomials in the form (a + b)(a - b) = a^2 + b^2 or learning combining the step of multiply by -1 with another action, like moving things from left to right.

I hope that helps.

Dang. I tried to use the FOIL method but somehow it wasn’t adding up. Guess my math brain needs some more working on… Thanks for your time. It makes total sense now!

And on the second problem the multiplying by -1 is what I missed. (But shouldn’t it be “-25 + u^2 = -16” after multiplying by -1?)

1 Like

Yes it should - good spot! That was a typo on my part and proof that it happens to all of us! :smiley:
(I’ve now edited the original response).

1 Like

Right on… My math brain starts engaging. :smiley:

1 Like

For y = -1/3x^2 + 10/3x - 16/3:

-3y = x^2-10x+16
(5+u) (5-u) = 16
u^2 = 25 - 16
u^2 = 9
u = 3

y = -1/3 (x-8) (x-2)

1 Like

the answer is y=(-1/3)x^2+(10/3)-(16/3)=(-1/3)(x-8)(x-2)

1 Like