Mindblown: a blog about philosophy.

Factors Factors refer to numbers that can be multiplied together to reach a specific product. For example, 2 and 3 are factors of 6 because their product is 6. Quadratic factors were discussed above, while this section will focus on whole number/constant factors. These types of factors usually come up on the ACT in the following…

Probability with “not” Probability questions with not are similar to normal probability questions. The main difference is that the odds that need to be calculated are the odds of failure rather than success. In the earlier examples on probability, the focus was on drawing a specific color of starburst out of a bag. The odds there were…

Similar triangles Similar triangles are triangles that have the same angles, but different side lengths. The easiest way to think of this is to imagine a triangle being scaled up or scaled down. If specific numbers aren’t given for a similar triangles problem, then it’s best to look at the shape of the triangle. If the original…

Change the base – If 9x = 275, what is x? Some questions will require changing the base of a number with an exponent. For example, 16x = 2564, then what is x? 256 = 162 , meaning that the original equation can be rewritten as 16x = (162)4 = 168 This will usually involve numbers whose square or cube roots are simple integers. So it’s useful…

Triangle opposite side rule The sides of triangles and the angles opposite them are related and this relationship may be useful on certain triangle questions. The most important thing to remember is that the length of the side of a triangle is proportional to the angle opposite that angle. So the largest side will also have…

Laws of Sines and Cosines The Laws of Sines and Cosines are useful for geometry questions involving non-right triangles. Law of Sines The Law of Sines states that  aSin(A)=bSin(B)=cSin(C) where a, b, and c are the side lengths of the triangle, and A, B, and C are the internal angles opposite those respective sides, respectively. Questions involving the Law of…

Arithmetic sequence Arithmetic sequences are similar to function, in that they indicate a way to work with numbers. However, instead of working with numbers continuously (including terms with decimals), sequences work only with whole numbers. Questions will normally provide a sequence, which students will then have to determine a specific term in (for example, the…

Difference of two squares Difference of two squares is a pattern that can be used to factor certain quadratic equations on the ACT math section. The default formula is that: a2 − b2 = (a+b)(a−b). The terms in this formula can be either variables or constants. There is no center term because the two binomials are conjugates of each other. When…

Circle equations The default equation of a circle looks like this: (x−h)2 + (y−k)2 = r2 where r is the radius of the circle, and the x and y binomials indicate where the center of the circle is. Circle questions on the ACT math section will normally provide either a graph or an equation. Whichever one is provided must be used…

c = product of roots, b = sum of roots, a = scaling When factoring an exponential equation (ax2+bx+c into (x-d)(x-e)), the signs and magnitudesof b and c can provide a lot of information that is useful for factoring. Example Equation: x2 + 3x − 10 C C should always be the starting place any time a student factors. It gives direct and specific information…