Quadratic skills

There are four crucial skills a student needs to have when it comes to quadratic equations: to factor, foil, zero parentheses, and graph parabolas.

To start, it’s worth remembering what a typical quadratic equation looks like:

ax² + bx + c

There are three terms: a quadratic (squared) term, a linear term, and a constant.

• Factor

Quadratics are often expressed in the above format, but the ax² + bx + c form isn’t necessarily super intuitive. Oftentimes, it’s helpful to factor a quadratic equation to make it easier to interpret. Factoring a quadratic means the same thing as factoring a normal number: looking for things that can be multiplied together to get the original number/expression.

To factor the number 27, one can either factor it by expressing it as 27 x 1, or as 9 x 3. The idea is the exact same with quadratics, but now there are both whole numbers and variables to work with.

Quadratic equations will often factor into two binomials (an expression with two terms that are either added or subtracted from one another). Binomials look like: (x + a) or (x−b).

When looking to factor ax² + bx + c, one is first going to look at the aand c terms. That’s because if the quadratic equation factors cleanly, these two numbers will be the product of only two things.

ax²comes from the two x terms multiplied together. c comes from the two constant terms (a and b) multiplied together. Since these two terms are only the product of two numbers, it gives a student the most direct information about what the factors might be.

One usually starts with the c term, since it’s the most basic. C is just the product of two constants. So C is factored like any other integer.

The example quadratic here will be:

x² + 5x + 6

c = 6. The factor pairs of 6 are; 2 and 3; 1 and 6. The binomial factors need to include one of these two pairs.

a = 1. The leading coefficient on the quadratic term is 1, so there are no coefficients in front of the variables in the binomials. The middle term products are also not scaled by anything.

Since the sign on c is positive, the two constants inside the binomials need to have the same sign. Since the middle term b is positive, it’s clear that the constant terms will also be positive.

This equation requires two binomials with no coefficients on the x, which multiply to 6 and add up to 5. Therefore, the factor pair of the constants is 3 and 2.

So factored, the above equation would look like: (x+3)(x+2)

• FOIL

FOIL stands for: First, Outside, Inside, and Last. It’s a way to remember how to multiply out two binomials.

The first two terms of each binomial are multiplied together. Then the numbers on the outside of the expression as a whole, followed by the two numbers on the inside of the expression as a whole. Finally, multiply together the last two numbers from each binomial.

These four things are then added together to tell one what the total product of these two binomials is equal to. The First and Last terms normally stand on their own, since they are usually the products of two variables and two constants, respectively. In comparison, the Outside and Inside terms can usually be added or subtracted together, since they will usually both be the product of a variable and a constant.

Check out the explanation in #55 on conjugates below to see why they create quadratic equations with just two terms when FOILed!

• Set parenthesis equal to zero

Part of the use of factoring quadratic equations into binomials with variables and constants is that it can make key information about the equation much easier to see. For example, when equations are factored into binomials such as (x−5)(x+3), the x-intercepts of the graph are much easier to find than when the equation is expressed as x² − 2x − 15.

To find the x-intercepts, simply set each parenthetical pair = 0. Another way of thinking of it is to ask: what number would need to be substituted for x to set the parentheses to 0. The answer should always have the opposite sign from the sign in between the two binomial terms. If there is a minus sign, then the answer should be positive. If there is a plus sign, then the answer will be negative.

• Graph parabolas

Students should be able to look at a quadratic equation and immediately know what the graph of that equation should look like.

For example, the equation: y =  − (x+1)² + 2

The main things to look at when trying to graph a parabola are:

Leading Coefficient

• If the leading coefficient is positive then the parabola should be upward facing and the parabola will have a minimum point

If the leading coefficient is negative then the parabola should be downward facing and the parabola will have a maximum point

In the example above, the leading coefficient is -1, meaning that the parabola is downward facing and has a maximum point.

Intercepts

The intercepts show where the parabola will hit its maximum or minimum point.

• The x-value of this minimum point is equal to the solution to the parabola’s binomial. In the sample equation above this solution = -1
• The y-value of the minimum point is the value of y at the solution point. For the equation above y=  − (1−1)² + 2 = 2
• Combining these two things shows that the maximum point for the parabola should be at (-1,2)

Equations with Multiple Quadratics

Not all equations will have only one quadratic in them. For example, a problem could ask about the graph of: (x−3)²(x+2)³

There are two key things to look at here:

• Exponent: The exponent says what the equation will do at the given intercept.
  • If the exponent is linear (raised to the first power), then the function will cross the x-axis at that intercept
  • If the exponent is quadratic (raised to the second power), then the function will “bounce” off the x-axis at that intercept
  • If the exponent is cubic (raised to the third power), then the function will “slide” at the x-axis (it will cross, but not in a linear fashion. Steeper further from the intercept and shallower close to the intercept)
• Intercept: The factors of the equation show how many x-intercepts the graph should have.
  • There will be a unique intercept for every unique value of x that could set the equation to 0. This is usually found by looking at the constants in the binomials and seeing if there are any numbers repeated in the parenthesis and which have the same sign. If the sign and constant are repeated, then this does not create an additional intercept.

• “Real Solutions” with Quadratic Equations

When a question presents a quadratic equation and asks about “real solutions” or simply “solutions”, there is a two step process to finding the answer the question is looking for.

Step 1: Try to Factor

See if the quadratic equation factors cleanly. If the quadratic equation can be factor into two binomials with whole numbers, then that will always be the quickest and easiest way to find “solutions” to that equation. Because this is the easier way to solve quadratic equations, questions that are designed for this method will often come earlier in the math sections. They also will often be the problems that mention just “solutions” rather than “real solutions”. Non-real solutions tend to show up more during the second step.

Step 2: Look at the Discriminant

−b±2a For these questions, it’s necessary to use the quadratic equation. The most important portion of that equation is called “the discriminant”, which refers to the portion of the equation that is underneath the square root: b² − 4ac.

The discriminant determines if there is a negative, positive, or zero number underneath the square root. This then determines how many “real” answers there are to the equation.

A negative number under a square root means that there are only non-real answers (answers that include i (. Zero under the square root means that the equation simplest to just 

−b2a, which since a and b are constants, which will lead to just one answer. And finally a positive under the square root leads to a positive number, which will then be added and subtracted from  − b, which then creates two answers.

Discriminant	# of Real Answers
b2 = 4ac	1
b2 < 4ac	0
b2 > 4ac	2

Just like with linear equations, questions about real answers for quadratic equations will either ask a student to find how many real answers there are for a given equation (unlike with linear equations, there is just one quadratic equation, since quadratics can have multiple solutions on their own) or they will ask a student to create an equation that either CAN or CANNOT be true if the equation is to have a given number of real solutions.