Quadratic equations have three terms and are in the form ax² + bx + c. An example of a quadratic is x² – 5x + 6. To find the factors of this equation, we must set up our set of two parentheses: ( )( )
The first term in both parentheses must be x, since x multiplied by x is the only way to get x². Then we look at the coefficient of the second term, -5. It’s important to include the sign in front of the integer as part of the coefficient. One of the rules of quadratic equations is that the second terms in the two factors must add together to equal the middle term’s coefficient. So we need to think of two numbers that add together to give us -5.
Already, we can think of many combinations: -6 and 1, -2 and -3, -200 and 105. So which pair is it? Now we have to look at the integer that’s the third term of the quadratic. Here it’s + 6. Another rule of quadratic equations is that the third term of the quadratic equation will equal the product of the second terms in the two factors. So not only do we need the two numbers to add together to equal -5, but we need them to multiply together to equal + 6. Therefore the factors must be: (x – 2) (x – 3). The “roots” or the “solutions” for this quadratic would be 2 and 3.
Systems of Equations
The ACT will often present you with two or more equations with multiple variables. Remember the “n equations with n variables rule.” If there are 2 variables in an equation (for example, x and y), then there must be 2 equations that each contain those variables in order to solve. The two common ways to solve are Substitution and Combination. Our Experts review each method in detail here.
It’s helpful to think of (x, f(x)) as another way of writing (x, y). For many function questions, you can Pick Numbers for the variables to solve!
For which of the following functions f is f(x) = f(1 – x) for all x?
A) f(x) = 1 – x
B) f(x) = 1 – x2
C) f(x) = x2 – (1 – x)2
D) f(x) = x2(1 – x)2
E) f(x) = x / 1-x
For instance, with here f(x) = f(1-x), if we pick x = 4 and plug in, we can rewrite the function as f(4)=f(-3). So for the answer choices, we could simply replace each function with 4 and -3, and see if f(4)=f(-3).
- F(-3) = 1 – (-3) = 4
F(4) = 1 – (4) = -3
They are NOT equal. Eliminate. Continue this method with the other choices, until you find the function for which the value you get when you plug in 4 is the same as when you plug in –3. That choice is D:
- F(-3) = (-3)2 (1 – (-3)2 = (9)(16)
F(4) = (4)2 (1 – (4))2 = (16)(9)
F(-3) = F(4). This is the correct answer.
Logarithms are a unique way of writing exponents. We’re used to seeing exponents in a format like y = xa. In “logs” that equation is equal to logx(y) = a. This is the most essential piece of information you’ll need to solve logarithms. If you want to memorize all the possible logarithm rules, check out this full list here.
This concept sounds challenging, but really only requires basic addition and subtraction, skills you already have! The key is to add and subtract the corresponding elements, row by row, and column by column. For a full example of this, check out this post.