[Education] What is important in junior high and high school mathematics -> Knowing multiple ways to solve a single problem and working on them until you can reproduce them all yourself [With problem examples] [One of the vast number of elements you need to know to turn study into scores]
A certain math problem was posted on X (formerly Twitter), and even though I was a student 30 years ago, I thought about it.
Problem below. What do you think?
There seem to be at least three ways to solve the first question of this major problem just by thinking about it for a moment. I came up with them too.
Spoiler alert below
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1. Brute-force calculation method.
Since α is the positive solution to x^2 - 5x - 1 = 0,
find the positive solution using the quadratic formula.
Then, substitute that solution into the expression to be found,
α - 1/α, and calculate diligently.
2. Use the relationship between the roots and coefficients of a quadratic equation.
Let α be the positive solution to x^2 - 5x - 1 = 0, and let β be the other solution. From the relationship between roots and coefficients,
α + β = 5 and αβ = -1.
Since it can be transformed into β = -1/α,
substitute this into α + β to get
α + β = α - 1/α = 5.
The answer is found.
3. Degree reduction.
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(Point 1) The problem can be transformed into α - 1/α = (α^2 - 1)/α.
(Point 2) Looking at the original equation, I thought, 'Oh, it looks like I can make α^2 - 1 into an expression of only α' (in other words, I can turn a squared expression into a first-degree expression = degree reduction).
Thinking from these two points...
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Since α is a solution to the equation, α can be substituted for x.
Even if you substitute α for x,
α^2 - 5α - 1 = 0 holds true.
Transform this to get α^2 - 1 = 5α.
Here, transform α - 1/α = (α^2 - 1)/α.
Substitute α^2 - 1 = 5α into this expression,
and since α ≠ 0,
α - 1/α = 5α/α = 5.
The answer is found.
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Now, the important thing is to work on all three methods until you can reproduce them with your own hands. This will save you during the actual exam if you start solving with one method and get stuck.
If you only know one way to solve it, you get zero points if you get stuck. You should try to work on alternative solutions in your daily practice.