Hopefully you were able to find some time and a quiet place this weekend to try our Five Problems for Your Weekend (3/27/15). Below you’ll find the fully worked-out solutions to those problems.

7. D

Carefully “FOIL” out the problem paying close attention to signs

(3*x* – 4)(*x* + 5)

3*x**² + *15*x* – 4*x* – 20

3*x**² + *11*x* – 20

22. F

Begin by getting zero on one side of the equation. Then factor and use the Zero Product Property. This is another problem where you need to be *very* careful with signs.

*x**² + *10*x* = 24

*x*² + 10*x* – 24 = 0

(*x* + 12)(*x* – 2) = 0

*x* = -12 and *x* = 2

35. C

There is a formula for finding arc length: *s* = θ*r*. In order to use this formula, you need to convert the angle in *degrees* to an angle in *radians* by multiplying by π/180. For the angle in this problem that calculation would look like this: 30(π/180) = π/6. To find the arc length you can multiply that angle by the radius of the circle. For our arc, that would be (π/6)(6) = π.

But what if you don’t remember the arc length formula? You could try more of a logical approach using what you know about the circumference of the circle. For this circle, the circumference would be *C* = 2π*r* = 2π(6) = 12π. The arc length we want is some fraction of that total circumference. But what fraction? Recall that a full circle is 360 degrees so our arc is 30/360 or 1/12 of the entire circumference. Multiplying 12π (our circumference) by 1/12 gets you an arc length of π, the same answer we got using the formula.

44. J

Recognize that squaring a complex number is very much like squaring a binomial; it’s really a “FOIL problem in disguise.”

(6 – *i*)²

(6 – *i*)(6 – *i*)

36 – 6*i *– 6*i + i²
*36 – 12

*i*+ (-1)

35 – 12

*i*

You need to use the fact that *i*² = -1, but that fact is almost always (I think I really want to say *always*) given to you on the test.

55. B

For every value of *x*, it is true that *f*(-*x*) = – *f*(*x*). This is another way of saying “opposite *x*‘s give you opposite *y*‘s. If you look at π, for instance, you’ll see that *f*(π) = 3. Now go look at -π and you’ll see that *f*(-π) = -3. This function is *odd*.

If you have questions about these problems or anything else to do with the ACT, leave a comment below or send me an email at info@cardinalec.com.

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