With factoring, completing the square, quadratic formula, and the square root method, it can be difficult to decide which method is most efficient in solving quadratic equations.
Recall in my last post, I introduced a girl who was deciding whether a private math tutor or a tutoring center would be a better fit for her to learn Algebra I. This is the math course most freshmen take in high school.
At our first session, Emily amazed me with how quickly she learned new problem-solving methods. She had never seen a system of equations but she devised a way to solve it! Her mathematical intuition left me wide-eyed, stumbling over my words. I could barely contain my excitement (never underestimate what someone can do!).
I tutored two other Algebra I students that year. They had Algebra class five days a week along with tutoring with me once or twice a week. Emily was in Pre-Algebra in school, so she only learned Algebra one hour a week with me.
After one month of tutoring, Emily was ahead of the other Algebra students. With only tutoring once a week. The personalization of private tutoring, focusing only on topics she needed to further develop, allowed for us to cover a full week’s worth of content in one session.
Not only was the pace quick, but her test scores proved she mastered the content. I gave her the same tests I gave my former Algebra I classes. On each test she earned an A.
By December, we slowed down the pace, but we were still halfway through the curriculum. We were able to keep up the pace of a normal Algebra class meeting only one hour a week.
We finished the Algebra curriculum in April and began to review for her Algebra placement exam.
Even Advanced Students Encounter Challenges
My stomach dropped as I read an email from Emily’s mom:
“Emily was pretty upset about the review when she came home today. She felt like she forgot everything…Do you think she understands most of the concepts in Algebra 1? I don't want to push her ahead if she really isn't ready.”
If Emily wasn’t prepared for the placement exam, I blamed myself. I should have reviewed concepts more consistently throughout the year.
During that previous review session, I could see Emily’s discouragement as she aggressively erased wrong answers. Her shoulders slouched more with every question she couldn’t correctly solve.
Math came naturally to her and she was used to excelling in it. She had high expectations of her math abilities and she was frustrated. It was the first time in her life that she encountered math topics she didn’t fully understand.
During that review session, I suppressed my fear that she hadn’t retained anything I taught her. As soon as I gave her a clue word or the first step, she had an "Oh yeah!" response. The information was all there, she needed to review to remind herself.
Between sessions, I encouraged her to review on her own. I taught her how to study math, to look back through one chapter of session notes each week.
I gave her mini quizzes at the next tutoring session. She didn’t need as much guidance solving problems on the quiz. With each correct answer, I regained hope. Seeing a glimpse of success boosted confidence for both Emily and me.
How to Solve Quadratic Equations
After 5 more tutoring sessions, I gave her a post-assessment. The results were greatly improved in most topics, but she still hadn’t mastered quadratics. I provided Emily with videos and practice problems to study for her placement exam. I also created the diagram below to help her connect solving quadratic equations using different ways.
There are more analytical methods for solving quadratic equations other than factoring. But factoring is the most common and can be challenging for students. Learn how to factor every type of quadratic expression.
But which Method is most Efficient?
1. Factoring—most popular method, but not always possible.
This is the Analytical method I used in the picture above. Since we have a trinomial with 3 terms, in x^2+bx+c form, we need a pair of numbers that multiply to c and add to b. So for this example, we need a pair of numbers that multiply to -2 and add to 1. That would be 2 and -1 since 2*-1=-2 and 2+-1=1. The 2 and -1 become each of the factors in parentheses. Once your quadratic is factored, set each factor equal to 0 and solve separately. This will give you the solution(s) to the quadratic equation. You can have 0, 1, or 2 solutions to the quadratic equation.
2. Quadratic Formula—always works but can be a long process. Below I worked out the same quadratic equation using quadratic formula.
3. Completing the Square—if "b" is even. I typically avoid this method unless the directions specify to use this method because it's the most complicated. I find the other methods to be more efficient and students find the solutions using other methods more accurately.
4. Square Root method—quickest method but can only be used if there's no "b" term. For example, if you had:
The common mistake I see with this method is students forgetting that when you take the square root to solve an equation, you have both 3 and -3 as solutions. That's because 3^2=(3)(3)=9 and (-3)^2=(-3)(-3)=9. Both 3 and -3 make the equation true.
"Fun" fact (I find it fun at least!): that problem could also be solved by factoring! It's a difference of squares so x^2-9 could be factored to (x+3)(x-3) and setting each of those to 0 would also give you -3 and 3 as solutions.
To dive deeper into those methods, watch a video explaining more methods for solving quadratic equations. In the video, I provided example problems demonstrating how to use each method.
See more step-by-step examples of solving quadratic equations.
Struggling as a Growth Opportunity
A month later I heard back from Emily’s mom. She placed out of Algebra I, performing “better than most current Algebra 2 students” would on the test.
Entering high school can be a scary time for both parents and students. Some students will easily ace every test. Others, even advanced students, will encounter a topic or subject that is difficult for them to grasp. Whether the struggle is in solving quadratic equations or not until differential equations in Calculus, they don’t have to go through the frustration of not understanding alone.
At Meryl’s Magic Math, we pride ourselves in explaining concepts as simply as possible and in multiple ways to ensure understanding. In tailoring the lesson to the student’s specific needs, we are able to work together to cover a full week’s worth of class content (and sometimes more!) in one hour. Fill out our New Student Form to get your child started with individual math tutoring.