Quadratic Equations

The General Form of the EquationA quadratic equation is a second degree polynomial, usually written in general form:

𝑎𝑥2+𝑏𝑥+𝑐

The a, b, and c terms are called the coefficients of the equation, and the x guy is the variable.

Technically, the a coefficient is called the quadratic coefficient, the b is the linear coefficient, and the c is the constant term.

For whatever reason, these equations are called quadratics, even though they’re always a second degree polynomial (“quad” means “four”).

Solving Quadratic EquationsA solution to a quadratic equation is an x value such that the following equation is true:

𝑎𝑥2+𝑏𝑥+𝑐=0

Such an x is called a root of the polynomial.

Fortunately, there’s a formula, cleverly referred to as the quadratic formula, that allows us to find these roots.

Here it is:

𝑥=−𝑏±√𝑏2 − 4𝑎𝑐2𝑎

The ± term lets us know that there are two roots to this polynomial. This makes sense, since we have an term in our equation.

Quick ExampleFind the roots of the quadratic equation: .

The most difficult part about solving quadratic equations is remembering the quadratic formula; from there, it’s just a matter of plugging in numbers into it.

Here it is again!

𝑥=−𝑏±√𝑏2 − 4𝑎𝑐2𝑎

In this example, and .

(Even though it doesn’t explicitly say “1x2 + …” in the equation, the 1 is understood; it’s just like in grammar, when commands such as “get out” have an understood “you” in front: “(You) get out”).

Let’s plug in our a, b, and c values:

𝑥=−3±√32 − 4 (1)(2)

2(1)

Time to simplify!

𝑥=−3±√32 − 4 (1)(2)

2(1)

𝑥=−3 ±√9− 82

𝑥=−3±√12

𝑥=−3 ± 1

2

If we choose + in our ± part:

Then choosing – givesus

This will give us two answers, since the ± symbol means we have a choice between + and – .

So, our roots of the quadratic are and .

We can check that these are indeed the roots by plugging them back into our equation:

:(−1)2+3 (−1)+2¿1 −3+2¿− 2+2=0

:(− 2)2+3 (− 2)+2¿ 4− 6+2¿− 2+2=0

So are the roots to the quadratic equation .

A Special Name for a Certain Type of QuadraticEquation That’s Not as Really as Special

In that example, the quadratic coefficient a was 1.

Quadratic equations like that are called monic polynomials.

Whether a quadratic equation is monic or not isn’t really a big deal, though it can come in handy if all the coefficients are divisible by the quadratic coefficient.

What is really neat, though, is that any quadratic equation can be written as a monic one!

𝑎𝑥2+𝑏𝑥+𝑐All we have to do is divide everything by a:

𝑎𝑥2+𝑏𝑥+𝑐a

𝑎𝑎𝑥2+

𝑏𝑎𝑥+

𝑐𝑎

𝑥2+𝑏𝑎𝑥+

𝑐𝑎

This is really only needed to make the quadratic formula easier to solve, since in this case we have one less variable to plug in. Sometimes, it makes it harder, though, so you have to be careful!

The DiscriminantThe discriminant of the quadratic formula tells us a lot about the number of roots the quadratic will have.

The discriminant is this:

If our discriminant is positive:

We will have two distinct roots:

If our discriminant is zero:

We will have one root:

If our discriminant is negative:We won’t have any real roots!

Side note: if the roots are perfect squares (meaning the discriminant is of the form a2), then our roots are rational numbers (fractions or whole numbers).

𝑏2− 4𝑎𝑐

and

𝑥=−𝑏2𝑎

Factoring QuadraticsWhen we know the roots of a quadratic equation, we are able to factor it.

If the roots to a quadratic are u and v, then the quadratic can be factored into: (x – u)(x – v)This is something that “just is”. It makes sense mathematically, though it’s one of those things that are never derived.

If we were to plug in u into the above expression, then we would end up with:(u – u)(u – v) = 0(u – v)= 0The same thing would happen if we plug in v:(v – u)(v – v) = (v – u)0= 0

This is fantastic, since we know a number is a root of a quadratic when plugging it in gives us zero. Yay!

Quick ExampleFactor the quadratic equation: 2x2 – 4x – 6 = 0

In order to factor this guy, we need to find out what its roots are. That’s not bad; we just have to plug all the numbers into the quadratic formula and we’re done!

But, all the coefficients of the equation are divisible by 2. In this case, it’s much easier to divide both sides of the equation by 2, and then plug in what’s left into the quadratic equation.

2x2 – 4x – 6 = 02

Great! Now that we have this quadratic as a monic polynomial, we can solve for the x values that make this equation true.

22𝑥2−

42𝑥−

62=0

𝑥2−2 𝑥− 3=0

So, our roots are x = 3 and x = –1.

To factor this equation out, we just need to plug these numbers in as our u and v values from a couple slides ago , and then we’re home free.(x – u)(x – v)We can just put in 3 for u and –1 for v:(x – 3)(x – (-1)) = (x – 3)(x + 1)That’s it! Factored out, 2x2 – 4x – 6 = 2(x – 3)(x + 1).

𝑥=−𝑏±√𝑏2 − 4𝑎𝑐2𝑎

Time to plug the numbers into our quadratic formula:

Now our a is 1, b = -2, and c = -3:

𝑥=−(−2)±√(−2)2− 4 (1)(− 3)

2(1)

𝑥=2 ±√4+122

𝑥=2 ±√16¿ ¿2

𝑥=2 ± 4

2=

22

±42=1± 2¿ {3 if we use+¿−1 if we use−

Don’t forget the 2 we factored out before!

Completing the SquareCompleting the square is probably the most difficult thing to do involving quadratic equations.

After you complete the square, you will have something that will look like this:

k (x – a)2 + d

Where k and d are constants. Notice what happens when we multiply (x – a)2 out:(x – a)2 = (x – a) (x – a)= x2 – 2ax + a2

So, it appears that the middle b term is twice the number that appears in the completed –squares part.

Also, our c term is just that number squared.

When we’re out to complete the square, we have to look at the middle b term. That number will tell us everything we need to know about what the last number will be in order to complete the square.

Example: Say we want to complete the square for x2 + 4x + 3.

Our middle coefficient is 4. So, the number we’re looking for must be half that, so we want a 2:

(x – 2)2

But, the last term in our equation is 3. We know it’s supposed to be 2 squared, which is 4. What do we do about this?

There’s a nifty trick people like to use, where essentially we’re adding 0 to our expression (which of course is a perfectly legal move; adding zero to anything doesn’t change the expression at all).

But, instead of just adding 0, which won’t help much, we can rewrite 0 as 1 – 1.

So, our quadratic becomes x2 + 4x + 3= x2 + 4x + 3 + (1 – 1)

= (x2 + 4x + 3 + 1 )– 1

= (x2 + 4x + 4)– 1

= (x + 2)2 – 1

We’re done! That’s how you complete the square for any quadratic equation you like.

Why 1 – 1? I’ve chosen 1 because we need 4, but we’re only given a 3. What do we do to make 3 equal to 4? We add 1 to it! But, we have to subtract 1 also, otherwise we’d be changing the expression (which is an extremely bad thing to do).