Tutorial: Solving Quadratic Equations

What is a Quadratic Equation?

A quadratic equation is any equation (equation means that there is an equals sign) that has a degree 2 term and no higher degree terms in it. It might look like: $x^2+3x-10=0$ or $(x+1)^2 = 16$ or $(x+3)(x-4)=0$ (that one has a secret second degree term in it… you need to multiply the binomials to see it).

Do you need to brush up on your polynomial terminology?

What does it mean to solve a quadratic equation?

Any time you’re asked to solve an equation, you are being asked to find out what value or values the variable can be that make the equation true. For a simple equation, if $x^2 = 16$ , you solve it by taking the square root of both sides: $x = 4$ or $-4$ . Sometimes it’s called solving for x, sometimes you’re asked to find the roots of an equation, sometimes they’re called the zeros. There are a number of ways to solve quadratic equations; we’ll talk about three of them.

The Sledgehammer (the Quadratic Formula)

If a quadratic equation can be solved, the quadratic formula will get the job done. There might be an easier way to do it but the quadratic formula will definitely do the trick. Here’s the formula:

$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ (Which we derived here.)

What are all of those extra letters: a, b, and c? Those are the coefficients of the quadratic. The coefficient on the $x^2$ is a, the coefficient on x is b, and c is the constant term. In the equation $x^2+3x-10=0$ , $a=1$ , $b=3$ , and $c=-10$ . Now plug the values of a, b, and c into the quadratic formula to find the value(s) of x:

$x=\dfrac{-(3)\pm\sqrt{(3)^2-4(1)(-10)}}{2(1)}=\dfrac{-3\pm\sqrt{9+40}}{2} = \dfrac{-3\pm\sqrt{49}}{2} = \dfrac{-3\pm7}{2}$

When you finish calculating, $x=2$ or $x=-5$ . To check your answers you can plug each into the original equation and make sure that the result is true.

For $x=2$ : $x^2+3x-10=(2)^2+3(2)-10=4+6-10=0$ Great!

For $x=-5$ : $x^2+3x-10=(-5)^2+3(-5)-10=25-15-10=0$ Also works!

If you’re having trouble remembering the quadratic formula, I was taught to sing it to the tune of “Pop goes the Weasel.”

What if the equation is factored? ZPP to the rescue!

First, I want to make sure we’re talking about the same thing. I’m only considering an equation factored if the factors are on one side of an equals sign and the other side is zero. Something like this: $(x+3)(x+2)=5$ is not factored… that 5 won’t let you use ZPP! For an equation to be factored, you need something times something else equals zero.

What is ZPP? ZPP stands for the Zero Product Property. You can’t multiply two non-zero numbers together and end up with zero, so ZPP says that if two things multiply to zero, then at least one of those things must be equal to zero. If $(x+3)(x+2)=0$ , then ZPP says that either $x+3=0$ or $x+2=0$ . Solve each equation for x, and you find out that $x=-3$ or $x=-2$ .

If you have a factored quadrilateral in front of you, it will be much easier to solve for x using ZPP than to multiply everything out and then plug into the quadratic formula. Actually, once you’re comfortable with factoring, some quadratics in standard polynomial form will be significantly faster to solve with ZPP than with the quadratic formula.

The Square Root Method

Sometimes an equation has a squared expression in it, and then an extra number added or subtracted, like this: $(x-2)^2-49=0$ . These are great to solve; most of the work is already done! If the equation is currently set equal to zero, move the number over to the other side: $(x-2)^2-49=0$ becomes $(x-2)^2=49$ . Now take the square root of both sides. Remember! If you introduce a square root, you need to include a “plus or minus” sign: $x-2=\pm\sqrt{49}$ . Simplify the square root (if possible), then solve for x: $x-2=\pm7$ , $x=\pm7+2$ , so $x=9$ or $x=-5$ .

If you have a polynomial in standard form that you don’t feel like factoring, and you’re not excited about using the quadratic formula, consider completing the square and solving with the square root method.

Let me sum up

The quadratic formula will always find the values of x that make an equation true, and you get to sing! However, sometimes using the quadratic formula is overkill, and factoring and using ZPP or completing the square and using the square root method would be a better use of your time. It’s a good idea to practice all three methods so you’re ready to solve any quadratic equations that come your way. Knowing more than one method also helps you to check your work (though you can always plug your answers back into the original equation to make sure that it’s true).

Sometimes you’ll come across a quadratic equation that just can’t be factored, and then when you use the quadratic formula or the square root method you end up trying to take the square root of a negative number. Do not despair! You didn’t break math! However, if you end up with a negative number inside of a radical (square root sign), you can safely say that there are no real values for x that make the equation true. It means other things as well, but we’ll save that for another day.

Do you also have a quadratic formula song? What’s your preferred method of solving quadratic equations? Please let me know your thoughts and any questions you have by commenting below. Do you know someone who should see this post? Please share it!