Solving Quadratic Equations

In our last post, we discussed that the solutions to quadratic equations are the values of x where y equals zero, or where the graph crosses the x-axis.

So, now you’re probably wondering, “How do I solve quadratic equations?”

There are three standard approaches to solving quadratic equations, which we’ll outline below. For each method, we’ll discuss how to use it and when you may (or may not) want to apply the approach. Note that for each method, you’ll need to make sure your quadratic equation is first set to the standard format:

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1. Factor Method

If you were taught to FOIL*, you may think of this approach as the “reverse FOIL.” Essentially, with the factor method, you transform your expanded quadratic equation back to its original binomial** form. After that, you set each component equal to zero to identify the roots, which are the solutions to your equation.

Example: Factor Method

Solve for (x^2) + 5x + 4.

We start by setting up our binomials, using p and q to represent our unknown values.

(x + p)(x + q) = 0

If we were to FOIL, or expand, this equation, we know that p*z must equal 4, and p+q must equal 5. Using guess and check, the only two numbers that satisfy both requirements are 1 and 4. Now we can update our equation.

(x + 1)(x + 4) = 0

Since the binomials are multiplied together, the equation will equal zero whenever either of the binomials equal zero. We can see that this occurs at the following values of x:

x = -1, -4

Thus, the values above are the solutions to the quadratic equation (x^2) + 5x + 4.

When to Use: Factor Method

Since the factor method is essentially a guess-and-check exercise, it’s best to use this approach only when the factors are easy to spot. Otherwise, you’ll spend too much time guessing and checking multiples, when you could be moving on to another problem.

2. Quadratic Formula

If you don’t like to memorize formulas, you may not be inclined to use this approach. However, in this case, a little memorization will make solving these problems much, much easier.

To apply this approach, you first arrange your equation in the standard format for quadratic equations. Then you simply apply the below formula and you have your solutions!

Quadratic Formula

Quadratic Formula

Where a, b, and c correspond to the coefficients of the quadratic equation:

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Example: Quadratic Formula

Solve for (x^2) + 5x + 4.

Using 1 for a, 5 for b, and 4 for c, we apply the quadratic formula and simplify:

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Thus, we can confirm that the solutions are -1 and -4, which are the same values we got using the factoring method.

When to Use: Quadratic Equation

This method is computational in nature and takes the guesswork out of solutioning. It also works on all quadratic equations, so it’s a reliable choice. Use this method if you have a complicated quadratic equation to solve. However, if you have an easier equation (like the example we used), you may not want to spend the time applying this formula. In those cases, the factor method will suffice.

3. Completing the Square

Completing the square is an unusual but helpful method for solving quadratic equations. With this method, we “force” the equation into a perfect square, which we then factor and simplify to solve for x.

Example: Completing the Square

Solve for (x^2) + 5x + 4.

First, we first set our equation equal to zero and move c over to the right-hand side. We then add a placeholder on both sides of the equation for where we will “complete” the square.

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To “complete” the square, we take b, divide it in half, square that entire value, and add it to both sides. Why both sides? Well, because that’s algebra!

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Next, we factor the perfect square on the left-hand side, and simplify the right-hand side.

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From there, the rest is easy! Just keep simplifying and you’ll find your solutions.

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And, just like the other methods, our solutions here are -1 and -4.

When to Use: Completing the Square

Completing the square is not always the easiest way to solve a quadratic. So why would we choose this method? It turns out that completing the square will come in handy when you take more advanced math classes. So, while it’s an important tool to have, if you’re taking an exam and have the option to use whichever method you prefer, you may want to try another method to save yourself some time.

We just reviewed three methods for solving quadratic equations, and when you may or may not want to apply each of them. However, no matter which method you prefer to use, it’s important to practice all three in order to keep your mathematical toolkit in top shape.

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* FOIL stands for First, Outer, Inner, Last, which refers to the order you multiply when combining two binomials.

** Binomial in this context means the sum or difference of two terms.