Introduction to Quadratics
Quadratics are an important topic on the ACT® Math section that includes several subtopics: Factoring, the Quadratic Formula, and Parabolas. Understanding quadratics well can earn you many points, as they appear in various ways throughout the test.
So, what's a quadratic? It's just an equation with an x² term, usually along with an x term and a number at the end. For the ACT® Math, that's all you need to know about the definition.
Basically, a quadratic looks like this:
or
These two forms are actually equivalent - that's the main idea you need to grasp to answer quadratic questions on the ACT® Math.
Factoring / FOIL
The most common quadratic questions on the ACT® Math involve factoring. It's usually straightforward, but signed numbers can easily cost you points.
Let's break down the FOIL method with an example:
- First: Multiply the first terms in each set of parentheses.
- Outer: Multiply the outer terms.
- Inner: Multiply the inner terms.
- Last: Multiply the last terms.
Now, add all the results together:
Simplify by combining like terms:
One More Example
The ACT® might throw in an to spice things up. Here's how to handle it:
Example:
- First:
- Outer:
- Inner:
- Last: (remember, )
Combine the Results:
Perfect Squares
Perfect squares are a shortcut for certain quadratic expressions. Knowing these patterns can save you time on the ACT® Math:
- Difference of squares:
- Perfect square trinomial:
You don’t need to know either rule, completing the square or the difference of squares, for the ACT Math® section. You can always solve by using FOIL. However, if you already know those two rules, and you can recognize them on the ACT Math® section, you can save a little time. The ACT Math® often gives questions that can be solved easily with one of the two rules.
Parabolas
Parabolas are the U-shaped graphs of quadratic equations. They're important on the ACT® Math because they visually represent quadratic relationships.
The general equation of a parabola is:
where is the vertex of the parabola.
The ACT® Math mostly cares about "zeros" - the x-values where the parabola crosses the x-axis (when y = 0).
Zeros are at and .
The test also cares about whether a parabola opens upward or downward:
This parabola opens downwards.
This parabola opens upwards.
Quadratic Formula
The quadratic formula is a more advanced topic that sometimes appears in higher-scoring questions: x =
They might also want you to know generally that a vertex exists. The vertex is the uppermost point of a negative or the bottom-most point of a positive parabola.
Key things to remember: - The first term is always -b - The discriminant (under the square root) is
Further Study
Discriminant
The discriminant ( ) tells us about the roots of a quadratic equation:
- - 2 distinct real roots
- - 1 real root (repeated)
- - no real roots
Should you know this? If you’re above a 23 or so, it’s a good thing to learn. It won’t get you all the points, but it’s likely to give you some.
Polynomials
One less important but nevertheless somewhat important topic for the ACT Math® section is polynomials. For the purposes of the ACT Math® section, polynomials refer to equations with coefficients and variables that have a “degree” higher than 2. So basically, would be a line, would be a quadratic, whereas would be a polynomial for our purposes.
The ACT Math® section mostly cares about whether odd or even degrees make a polynomial’s graph have how many different changes.
What is the minimum number of degrees possible for the polynomial below?
A polynomial of degree can have, at most, turning points.
That means in order to have 3 turning points, this has to be at least a 4th-degree polynomial.
You can also just look at how many x values.
Key Takeaways
- Quadratics are equations with x² terms
- FOIL method helps expand factored quadratics
- Parabolas are U-shaped graphs of quadratic equations
- The quadratic formula solves any quadratic equation
- The discriminant tells us about the nature of a quadratic's roots