4/m + m = 5, m =? How Much ALGEBRA do you know?

How to solve a rational equation – quadratic equations and extraneous solutions.

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[alexmentes]

Thanks Math Man. I'm 80 years old and have an undergraduate degree in math (Princeton 1967). The field I went into didn't require much math. Now to keep my mind active I enjoy watching your videos. They're a fun reminder of a past journey.

[tomtke]

(4/m) + m = 5
m{(4/m) + m = 5}
4 + m^2 = 5m
m^2 - 5m + 4 = 0
-1 × -4 = 4
-1 - 4 = -5
(m - 1)(m - 4) = 0
sol.1 => m = 1
sol.2 => m = 4

Option d)

[topkatz]

Did it in my head.

D:{4, 1}

Multiply both sides by m you get 4 + m²= 5m

Subtracting 5m from both sides and re-arranging terms you get

m² - 5m + 4 = 0
Easy to factor this to yield

(m - 1)(m - 4) = 0
m - 1 = 0 >>>> m = 1
m - 4 = 0 >>>> m = 4

[panlomito]

Easy does it again... to lose the m in the denominator of 4/m just multiply the whole equation with m so
m . 4/m + m . m = m . 5 or m² - 5m + 4 being a nice quadratic equation that we can solve
with FACTORING or with the ABC - formula.

In this case factoring is the easiest way: (m-1) . (m-4) = 0 so m1 = 1 and m2 = 4

But as a lover of the abc-formula I would like to adress this solution too... (changing m into x)

y = f(x) = ax² + bx + c = 0 so a = 1 b = -5 c = 4

First take notice that the quadratic equation is a curve in the form of a parabola,
with a top down if a > 0 or a top up if a < 0
if a = 0 then the equation will be 0.x² + b.x + c = bx + c being a lineair equation

In this case a = 1 > 0 so this is a top down parabola !

Now there are three possible positions of the parabola in relation to the x-axis that we can can find with the
discriminator D = b² - 4ac
If D < 0 then the parabola is fully above or below the x-axis so no crossings with the x-axis
If D = 0 then the parabola will touch the x-axis with its top
If D > 0 then the parabola will cross the x-axis twice

In this case D = (-5)² - 4 . (1) . (4) = 25 - 16 = 9 -> positive so 2 crossings of the parabola throught the x-axis

The distance of the crossings relative to the symmetry-axis of the parabola (so left and right to the vertical line throught the top of the parabola), I will call it delta:
delta = SQR (D) /2a so in this case delta = SQR (9) / 2a = 3 / 2 . (1) = 3/2

And the x-coordinate of the top xtop = - b / 2a = - (-5) / 2 . (1) = 5/2

Now we can find the 2 crossings with x = xtop -/+ delta so
x1 = 5/2 - 3/2 = 2/2 = 1 and
x2 = 5/2 + 3/2 = 8/2 = 4


Tadaaa, that looks like a lot of work but sometimes it will be impossible to factor the equation, for example try to factor
x² + 3x - 11 = 0 good luck !

But the abc-formula works always, even with x² +3x - 11... And additionally giving more insight in the shape and position of the parabola.

To end this lecture I will determine the 3 point of the parabola with x1 = 1 xtop = 5/2 and x2 = 4:
y1 = f(x1) = (1)² - 5 . (1) + 4 = 1 - 5 + 4 = 0 so the first crossing is at (1, 0) nice test for y = f(x) = 0
ytop = f(xtop) = (5/2)² - 5 . (5/2) + 4 = 25/4 - 25/2 + 4 = 25/4 - 50/4 + 16/4 = - 9/4 so top(5/2, - 9/4)
y2 = f(x2) = (4)² - 5 . (4) + 4 = 16 - 20 + 4 = 0 so the second crossing is at (4, 0) again testing y = f(x) = 0

Met een vriendelijke groet aan Kleermaker1000 !

[mr.mxyzptlks]

As comments state, multiply by m and you have a quadratic equation. Just keep in mind, not needed in this case, that 4/m excludes m=0 as a solution. This could arise in another example.

[chrisdissanayake]

4 m
— + — = 5
m 1

4 + m^2 5
—————— = —-
m 1

4 + m^2 = 5m

m^2 -5m + 4 = 0

(m -4)(m-1) =0

If m-4=0,
m=4
OR
If m-1=0,
m=1

Answer: d) 4, 1

[giannaleoci]

m^2-5m+4=0
m=[5+-rq(25-16)]/2
m=[5+-3]/2
m1=4 m2=1

[hexbinoban]

4/m + m = 5
m/4(4/m + m ) = 5 (m/4) <=> 1 + (m^2)/4 = 5m/4
4(1 + (m^2)/4) = 5m/4 (4) <=> 4 + m^2 = 5m
m^2 - 5m + 4 = 0 ==> 4 product -4 * -1 = 4, sum -4 + (-1) = -5, {-1, -5}
(m - 4)(m - 1) = 0, m - 4 = 0 <=> m -4 + 4 = 0 + 4 => m = 4
m - 1 = 0 <=> m -1 + 1 = 0 + 1 => m = 1
m = {1, 4 } ----> Ans : d
Proof by substitution: 4/1 + 1 = 5 or 4/4 + 4 = 5

[stevenjohnson]

Simple plug in the answers on the right you eliminate every other set but d 4, 1

[paulfrank]

I didn't do any algebra. It was too easy to just plug the answers into the equation and do the simple math to see which one worked, and I'm lazy.

[dazartingstall]

4/m + m = 5
m² + 4 = 5m
m² − 5m + 4 = 0
(m − 4)(m − 1) = 0
m = 4, 1

[pandurangaraonimmagadda]

m sqr-5m+4=0 (m-4)(m-1)=0, m=1 and m=4, so answer is (d)

[panlomito]

Since there are stupid multiple choice answers (stupid because only one is right) there is no need to "solve" this equation with algebra. Just put the answer into the equation:
a. 4/8 + 8 = 8½ next
b. 4/6 + 6 = 6 2/3 next
c. 4.9 + 1/9 = 36 1/9 next
d. 4/4 + 4 = 5 and 4/1 +1 = 5 so BINGO
I know this is not the most elegant way to find the right answer but multiple choice asks for it. There is no need to give 4 possible answers because the solution is very easy to find with a little algebra.

[argonwheatbelly]

d. m={1, 4} in my head in six seconds. Fun one!

[raya.pawley]

Thank you. (m-4)(m-1) Therefore m=4, 1

[russelllomando]

got 4, 1 instantly logic and math thanks for the fun.

[carlossummers]

Oooops. Time to brush up on my algebra.

[theDemocraticway]

I could solve it the easy way. Since it was multiple choice, I discarded the first 3 answers. Otherwise just turn it into a quadratic and this simple

[mauriziograndi]

D) 4, 1
4/4=1+4=5
4/1=4+1=5
m=4, 1
Video unnecessary long

[fubaralakbar]

Looks like Pre-Algebra, then sucker-punches you with late Algebra I. Clever!