PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022
Classic quadratics
Quadratics
Passport to advanced math
LEARNING OBJECTIVE
After this lesson, you will be able to:
· Recognize the classic quadratics
To answer a question like this:
Which of the following expressions is equivalent to 36x4y6 − 4?
A. 6(x4y6 − 2)
B. −6(x2y3 + 2)
C. (6x2y2 − 2)(6x2y2 + 2)
D. (6x2y3 − 2)(6x2y3 + 2)
You need to know this:
Memorizing the following classic quadratics will save you time on test day:
·
o This is known as a “difference of squares” because it takes the form of one perfect square minus another perfect square.
·
·
You need to do this:
When you see a pattern that matches either the left or the right side of one of the above equations, simplify by substituting its equivalent form. For example, say you need to simplify the following:
You would substitute (a − b)(a − b) for the numerator and cancel to find that the expression simplifies to a − b:
Explanation:
The expression 36x4 y6 − 4 is a difference of perfect squares. The square root of 36x4 y6 is 6x2y3 and the square root of 4 is 2, so the correct factors are . Hence, (D) is correct.
Try on Your Own
Directions: Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.
HINT: What can you do to make (a − b)2 in Q6 easier to work with?
6. For all a and b, what is the product of (a − b)2 and (a + b)?
A. a2 − b2
B.a3 − b3
C.a3 − ab2 + a2b − b3
D. a3 − ab2 − a2b + b3
7. Which equation does NOT have a solution at x = −4?
A.
B.
C.
D.
HINT: One of the given equations in Q8 is a classic quadratic.
8. A rectangle has an area of If the width of the rectangle is , what is the length?
A.
B.
C.
D.
9. A rectangular prism has a height of x, a width of 7 less than the height, and a length 14 more than the width. What is the volume of the prism?
A.
B.
C.
D.
HINT: For Q10, is there any variable you can factor out from each term?
10. Which of the following expressions is equivalent to the expression above?
A.
B.
C.
D.