4/m + m = 5, m =? Do you have the algebra skills to solve this equation? Let’s see…

How to solve a rational equation.

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TC Math Notes

[jerry]

Multiplying through by m, and then collecting all terms on one side to give a standard quadratic equation seems by far the easiest approach to me.

[drzlecuti]

M= 4 or m = 1, solved by direct observation in about 15 seconds; but I'm not the audience for this nicely detailed approach. Thanks for the A+!

[nycmax]

Another easier way to factor a quadratic equation with three terms is: after you set the equation and made it equal to zero, look at the "b"coefficient and the "c" value. ( b and c). Then try to find two numbers whose sum gives the "b" coefficient, and whose product yields the "C" value.
Ex: we have m^2 - 5m + 4, =0 . Here - 4 and -1 added up to give - 5, the "b' coefficient, and multiply together to give 4, the "c" value. Thus, we write (m - 4) (m -1) = 0 . Solutions: (m - 4 =0 --> m = 4) ; and ( m -1 = 0 --> m = 1 ).
the solutions may vary based on the " c" value. Here if "c" was 6 instead of 4, as in M^2 - 5m + 6 the solutions would then be: m = 3 and m = 2. Because -3 + -2 = - 5 and -3 x -2 = 6.😂 But if there is no 2 numbers that can be added up and multiplied to give the "b" coefficient and the "c" value, you use another method such as the quadratic formula. However this method will not work for quadratic equations with higher power such as m^3 m^4, m^5 that has more than 2 solutions and also that has a coefficient in the 1st term such 3m^2, 4m^3, 2m^5, etc...

[KenFullman]

The things that went through my mind were:
It has to be an integer.
It has to be less than 5
It has to be a positive number
It has to be a factor of 4
The only possible wrong answer that fits these criteria is 2
Both the other factors of 4 (one and four) are possible correct results.

[gaildawson]

I really like your lessons but I’m not going to listen to 5 minute videos with 4 commercials. I’m sure I can find someone actually interested in teaching.

[fredseeley]

I LOVE YOUR POSITIVE ATTITUDE AND PHILOSOPHY THAT EVERYONE CAN LEARN TO SOLVE MATH PROBLEMS SUCCESSFULLY IF YOU HAVE AN EXCELLENT TEACHER!!

[kevinreist]

m=1. 4/1=4. Then, 4+1=5 Didn't try to write it out, solved it in my head by looking at the equation. However, after watching it, I can see m=4 would also work because 4/4= 1. Then, 1+4=5. Sometimes, the correct answer is there is no solution to a poorly expressed (ambiguous) equation.

[davidh]

I just stumbled on this video, after learning the Japanese method of how to preserve mullet ovaries for use in pasta dishes. Let me boast a bit first. I am a physician, board certified in Internal Medicine. It wasn't easy getting there. I am fluent in two foreign languages, and can make myself understood in two others. I was an honors student in high school, Deans List for 6 semesters in college. I am the child of a high school math chairman who also was an associate professor of mathematics, and who actually wrote the math portion of a GED text book, the brother of a college math major who went into optical physics and was too lazy to go into engineering. In other words, I ain't dumb and the genes ain't too bad either. Dad once described me as the single worst math student he ever had to tutor, and was even worse than my mother/his wife. Everything I do, I excel in. With a small exception. I passed Calc 1 with a C, and by the grace of the gods, was able to take a Pass (ie a D) in Calc II. Those were gait controlling courses for med school. For more than 30 years, I have been wanting to learn calculus, and actually UNDERSTAND it. I've been trying to do YouTube problems when I stumble across them. I got to the factoring of your equation, met up with the negatives, and decided to quit there and watch the video. From the medical standpoint, I am fascinated by my true inability. Granted I had some really awful math teachers throughout my school years. But I am convinced that physically, my brain is missing a chunk of tissue that allows for the comprehension of these abstract concepts. I can take an equation, and just go so far. I look at a differential equation and solution and draw a total blank going step by step through the solution - I simply cannot comprehend what the hell is going on, or where the next line came from. It has to be my physiology, because everyone around me can do this stuff, I can't. And yet I am multilingual (language is stored in different places in the brain; my Italian is not sitting in with my English), I can explain pathology to a child. So in my old age (early 70s now), I have a goal before I leave this sphere. Dad's book comes out tomorrow, and I'll be following you on line. I promise, I won't call to say "Sorry, I don't understand how you got there..." Thanks for listening. Dr H, MD. Very smart, not so smart, and very frustrated by my math skills, or lack thereof.

[m.h.]

Solution:



4/m + m = 5 |*m → test required on solutions to eliminate possible extraneous solutions
4 + m² = 5m |-5m
m² - 5m + 4 = 0
m = -(-5)/2 ± √((-5/2)² - 4)
m = 5/2 ± √(25/4 - 16/4)
m = 5/2 ± √(9/4)
m = 5/2 ± 3/2
m = (5 ± 3)/2

test m₁ = (5 + 3)/2 = 8/2 = 4
4/m + m = 5
4/4 + 4 = 5
1 + 4 = 5 → correct, m₁ = 4 is a solution

test m₂ = (5 - 3)/2 = 2/2 = 1
4/m + m = 5
4/1 + 1 = 5
4 + 1 = 5 → correct, m₂ = 1 is a solution

So the equation has two possible solutions: m₁ = 1 and m₂ = 4

[BlackhawkPilot]

This can also be solved by dividing both sides by m. (4/m)/m=(4/m)m= 4 one solution. Substitute the 4 and the result is 4+1=5m and now you have the second solution of m=1. No quadratic necessary.

[waltdill]

A lot of comments and examples here that are just compressed versions of what is actually demonstrated in the video. But these are different things. It is easy to see and plug in the correct answer(s) without much thought, but that is only because the problem is relatively apparent. While proof, the QED, is rigorous. Even a quadratic formula is self evident primarily because it is convenient for the easier solutions. But going beyond this to the "whys" and "hows" is another level of insight.
Just a thought. Math is a casual hobby or side interest for me. Philosophical study has taught me to appreciate a sufficient argument in addition to an acceptable solution.

[russellbonesteel]

Greetings Mr. John and class! I am a retired teacher and I always wondered where the quadradic formula came from. One Summer day I decided to "Solve" The general trinomial AX^2 + BX +c =0 by completing the square.... The Answer was the Quadradic formula! I was soooo ready to collect my field metal! Nutz! What? Some mathematics guy beat me to the punch! I found that later in a mid 1980s math text! :) I am still working on precalculus edging into beginning calculus (self study for now) Your Amazing videos really fill in my gaps of knowledge Thank You Sir!

[wijnandhijkoop]

My 14 year old grand daughter has a disastrous math teacher. Everyone is happy that I can help her. Your examples are great. This example is exactly what she needs right now.

[fredseeley]

I WISH I HAD YOUR TEACHING GUIDANCE BACK IN THE DAY!!!

[GFlCh]

4
--- + m = 5
m
•••

Multiply everything by m:
4 m
--- * --- + m * m = 5 * m
m 1

m
4 * --- + m² = 5m
m

4 + m² = 5m


Reorganize:
m² + 4 = 5m

Subtract [5m] from both sides:
m² - 5m + 4 = 5m - 5m

m² - 5m + 4 = 0


Factor:
(m - 4) • (m - 1) = 0

Check factors:

[m² - 1m - 4m + 4]


Answers: m = 4, m = 1

[richardmullins]

By trial and error, m = 4. Then I noticed that we can treat it as a quadratic equation and it also has m = 1 as an answer, which I had not noticed before.

[MrSteeler]

It's a Quadratic Equation ax^2 + bx + c. Multiply everything by m, m(4/m)+m*m=5m. 4+m^2=5m,
m^2-5m+4=0, (m-4)*(m-1) = 0, m-4=0 and m-1=0, m=4 and m=1. Substitute into the original equation to check your answers. Some answers may be true and some may be false. This was an easy example but if your teaching beginners they really need to see how to solve Quadratic Equation because without this method there is no way to solve tougher equations.

[bijoychakraborty]

I really love your videos and take respect for your consistency ✔️
Love from India ❣️

[pacificdune]

Multiply by m to give standard quadratic equation m^2 - 5m + 4 = 0. Then factor to get solutions 1, 4.

[paulw]

Getting off to a good start in math is so important; not just the fundamentals either - as important is giving a kid CONFIDENCE in their ability to learn.