ACT and SAT Math Tips: Classic Quadratic Forms

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I’ve said it before and I’ll say Bright varicolored abstract of white picket fence by hydrangea shrubs with pink flowers and green foliage, for themes of summer, gardening, or landscapingit again: the writers of standardized tests assess the same concepts, over and over and over again.  Test experts are able to become test experts for that very reason—there is a finite amount of material one needs to learn in order to master the ACT and the SAT. If you study a few dozen tests and find these predictable patterns, you could be an expert, too.

Luckily, I’m here to save you from that mind-numbing task. And trust me—it’s mind-numbing, especially when we get into some of the concepts from Algebra II, such as classic quadratic forms.
This week’s tip comes from the Math section and looks at three common quadratic forms on the ACT and SAT:

         1.            (x + y)(xy)  =  x2y2

                         Examples:

                        (t + 5)(t – 5)   =  t2 – 52   =   t2 – 25

                        (3a + b)(3ab)  =  (3a)2b2  =  9a2b2

                        y2 – 64  =  (y + 8)(y – 8)

                        36 – n=  (6 + n)(6 – n)

 

 

        2.            (x + y)2  =  x2 + 2xy + y2

                        Examples:

                        (t + 5)2  =  t2 + 2(t)(5) + 52  =  t2 + 10t + 25 

                        (3a + b)2  =  (3a)2 + 2(3a)(b) + b2  =  9a2 + 6ab + b2

                        (6 + n)2  = 62 + 2(6)(n) + n= 36 + 12n + n2

 

 

        3.            (xy)2  =  x2 – 2xy + y2

                        Examples:

                        (t – 5)2  =  t2 – 2(t)(5) + 52  =  t2 – 10t + 25 

                        (3ab)2  =  (3a)2 – 2(3a)(b) + b2  =  9a2 – 6ab + b2

                        (6 – n)2   = 62 – 2(6)(n) + n= 36 – 12n + n2

 

Now let’s look at how memorizing these forms can help you quickly and confidently solve an ACT or SAT question.

 

1.     If rp = 7, what is the value of r2 – 2rp + p2?

        (A)     7

        (B)     14

        (C)     28

        (D)     35

        (E)     49

 

A student who doesn’t know the classic ACT and SAT quadratic equations will start to (unsuccessfully) use substitution:

r – p = 7

r = 7 + p

They will plug 7 + p into the second expression for r, even though the second expression is just an expression—not an equation!

r2 – 2rp + p2    

(7 + p)2 – 2(7 + p)p + p2    

(7 + p) (7 + p) – 2(7 + p)p + p2    

49 + 14p + 2p2 – 2(7p + p2) + p2        

49 + 14p + 2p2 – 14p – 2p2 + p2        

49 + 2p2– 2p2 + p2        

49 + p2        

Oh, wait! It’s just an expression. It doesn’t equal anything. This won’t work!

A lot of wasted time for the average student.

But for you, the reader of great test prep blogs, this question is solved in about 5 seconds. You look at r2 – 2rp + p2 and immediately recognize the third classic quadratic form:

(xy)2  =  x2 – 2xy + y2

So (r – p)2 =  r2 – 2rp + p2  

And since r – p = 7, then (r – p)2 =  72 = 49.

Voila.

Need more great tips? Check out our ACT classes.

 

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