A Better Way to Solve a Quadratic Equation

Hello everyone. Here is an alternate(somewhat simpler) method of solving a quadratic equation. I learnt it from a youtube stream by "3 Blue 1 Brown". The method was originally given by Po Shen Loh.

The Method is Below:

We all know that the equation of the form ax2+bx+c=0ax^{2}+bx+c=0 is known as a quadratic equation and the formula for its roots is b±b24ac2a\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}.
But there is a shorter formula which is much easier to use while solving problems. You just have to know the three facts below and then after a little manipulation, you are done!

Fact 1: sum of roots is ba\frac{-b}{a}
Fact 2: product of roots is ca\frac{c}{a}
Fact 3: difference of squares i. e. (md)(m+d)=m2d2(m-d)(m+d)=m^{2}-d^{2}

Let's introduce 2 new variables:
1) d=d = difference between the 2 roots . i . e ab|a-b|
2) m=m = mean of the 2 roots . i . e a+b2\frac{a+b}{2}
In the graph below, notice that αβ=(m+d)(md)=m2d2=ca\alpha\beta= (m+d)(m-d) = m^{2}-d^{2}=\frac{c}{a} (α(\alpha and β\beta are the 2 roots) because α=m+d\alpha = m+d and β=md\beta = m-d.
So d2=mp d^{2}= m - p where p is the product of the 2 roots.
Hence, the 2 roots are m±m2pm \pm \sqrt{m^{2}-p}

The formula for the roots of a quadratic equation m±m2pm \pm \sqrt{m^{2}-p} where p is the product of roots and m is the midpoint of the roots which can be derived from the 3 above facts.

Now, I will do an example of this: Solve the equation x2+3x+1=0x^{2}+3x+1=0

We have m=32m=\frac{-3}{2} and p=12p=\frac{1}{2}
Hence, by our above formula x=32±52x = \frac{-3}{2} \pm \frac{\sqrt{5}}{2}

The traditional quadratic formula can be derived from this and vice versa. If you have any questions or suggestions, please ask them in the comments. Try more examples to get the hang of it. Hope you liked this note and will start using this formula in the future!

Diagram is below

Note by Nitin Kumar
2 months, 2 weeks ago

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Lockdown Math series is so fun!

Adhiraj Dutta - 1 month, 3 weeks ago

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Except for the horrible time. I sleep, wake up and then sleep again as it's at 12. Lol

Nitin Kumar - 1 month, 3 weeks ago

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I just watch it after it ends. I can't concentrate when I'm sleepy. And my parents won't allow me to wake watch a livestream at midnight.

Adhiraj Dutta - 1 month, 3 weeks ago

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@Adhiraj Dutta Yeah, Grant should try to change the time as many Indians are watching the stream. Even I stopped watching them live as I am tired for the rest of the day.

Nitin Kumar - 1 month, 3 weeks ago

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Yup it is

Nitin Kumar - 1 month, 3 weeks ago

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Thanks, @Nitin Kumar. Really helps!

A Former Brilliant Member - 1 month, 4 weeks ago

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Also, proof of derivation would be extremely nice! I love this discussion!

A Former Brilliant Member - 1 month, 4 weeks ago

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what do you mean by "proof of derivation"?

Nitin Kumar - 1 month, 4 weeks ago

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Proof of derivation means : how did you derive your simplified quadratic formula from the quadratic formula? Also, still waiting for the graph!

A Former Brilliant Member - 1 month, 4 weeks ago

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@A Former Brilliant Member I put the graph! OK , Ill put the derivation.

Nitin Kumar - 1 month, 4 weeks ago

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Thanks for your complement!

Nitin Kumar - 1 month, 4 weeks ago

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Nice, @Nitin Kumar. Can you show the graph.

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ok, sure!

Nitin Kumar - 1 month, 4 weeks ago

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The graph is a general parabola.

Nitin Kumar - 2 months, 2 weeks ago

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