Quadratic equations have the form, ax2+bx+c=0, were and are given, and are sought. There are three traditional approaches to solving such equations: applying the quadratic formula, completing the square, and factoring. Unfortunately, though, students generally find it hard to remember the formula and the steps for completing the square, and they also find the trial-and-error aspect of factoring off-putting. But luckily, a new method has emerged that is easy to recall and involves no hit-or-miss guesswork.
In late 2019, Carnegie Mellon University professor Po-Shen Loh was brainstorming when, to his surprise, he stumbled on a simple and efficient new way to solve quadratic equations. “I was dumbfounded,” he declared. “How can it be that I’ve never seen this before … in any textbook?” And indeed, his new method is so straightforward and intuitive, it’s a minor miracle that it has escaped notice until now.
To illustrate Loh’s method and contrast it against the traditional ones, let’s apply all four to solving the equation x2-2x-48=0.
1.Solving x2-2x-48=0 Using the Quadratic Formula
The quadratic formula represents the solutions to the general quadratic equation ax2+bx+c=0 as
x= (-b±√(b2-4ac))/2a. Now you can see why students find this formula intimidating and hard to memorize! Anyway, for this problem, a=1,b=-2, and c= -48, so by the formula, the solutions x are
(-(-2)±√((-2)2-4(1)(-48)))/(2(1)) = (2±√196)/2 = (2±14)/2 = 1±7 = -6 or 8.
So, we arrived at an answer, but the demerits of the formula were on full display: It taxes memorization and forces complex and cumbersome computations.
2. Solving x2-2x-48=0 by Completing the Square
We will carry out the following four steps to “complete the square” for the given example.
A) Rewrite the equation as ax2+bx=c.
B) Add the quantity b2/4a to both sides.
Here, b2/4a=(-2)2/4(1) =1, so we have x2-2x+1=48+1; ie, x2-2x+1=49.
C) Express the left-hand side as a multiple of a perfect square.
(Note, Step B completed the left side as a perfect square, which therefore made Step C possible.)
D) Square root both sides of the re-expressed equation and solve for x.
√((x-1)2 )= √49
x= -6 or 8.
Observe that to use this method, students must recall and execute four intricate steps, which is something they find hard to do.
3. Solving x2-2x-48=0 by Factoring
In this approach, an attempt is made to factor the left-hand side of the given quadratic equation by searching for two numbers r and s whose sum is b and product is ac. So, here, we require r+s= -2 and rs=1(-48)= -48.
Now comes the trial-and-error part.
Does r=1 and s= -48 work? No, because while rs= -48,r+s-47 ≠ -2.
How about r=2 and s= -24? Still no, because while rs= -48,r+s-22 ≠ -2.
Several more tries and frustrating moments later, we find that the two numbers that work are r = 6 and s= -8.
Next, we use these two numbers in a process called “successive grouping” to arrive at a factorization of x2-2x-48.
x2-2x-48 = x2+6x-8x-48 (use the fact that b=r+s)
=x(x+6)-8(x+6)=(x-8)(x+6) (from successive grouping).
Last, we use the factored form of the equation to solve.
(x-8)(x+6)=0 ⇒ either x-8=0 or x+6=0 ⇒ either x=8 or x=6.
Closing note: Students generally find the search for r and s extremely exasperating, with good reason!
4. Solving x2-2x-48=0 using Professor Loh’s New Method
Let’s now experience the advantages that Dr. Loh’s method has over the three presented above. Here is his approach:
A) Compute the quantities M=(-b)/2a and N=c/a.
Here, M=(-(-2))/2(1) =1 and N=(-48)/1= -48.
Note, this step features two easy-to-remember and easy-to-perform calculations.
B) Solve the auxiliary equation (M+u)(M-u)=N for u.
Observe, the auxiliary equation in this step is easy to remember and easy to solve.
C) Quickly recover the solution to the original quadratic equation. They are, simply, the two factors in the auxiliary equation, M+u and M-u.
x=M+u=1+7=8 or x=M-u=1-7= -6.
Wow, that was almost too easy!
We have compared four methods for solving quadratic equations, three traditional ones, and a new one. We found that the new method, #4, was the fastest, easiest to remember, and easiest to perform. Therefore, students are advised to use it as their go-to technique whenever they must solve a quadratic equation.
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