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.
Carefully “FOIL” out the problem paying close attention to signs
(3x – 4)(x + 5)
3x² + 15x – 4x – 20
3x² + 11x – 20
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.
x² + 10x = 24
x² + 10x – 24 = 0
(x + 12)(x – 2) = 0
x = -12 and x = 2
There is a formula for finding arc length: s = θr. 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 = 2πr = 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.
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 – 6i – 6i + i²
36 – 12i + (-1)
35 – 12i
You need to use the fact that i² = -1, but that fact is almost always (I think I really want to say always) given to you on the test.
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 firstname.lastname@example.org.