Question

Is it possible to factor `y=x^2-7x+10`? If so, what are the factors?

Answer 1

`y = x^2 - 7x + 10 = (x - 5)(x - 2)`

Explanation:

There are different methods that can be used to factor your term. Let me show you one that is in my opinion the easiest for your question.

You are searching for a factorization like this:

`x^2 - 7x + 10 = (x + a)(x + b)`

This is true if you can find `a`, `b` so that

`{ (a + b = -7 ), (a times b = 10) :}`

It is easiest to start with the product.

Since `10` is positive, `a` and `b` need to have the same sign, so both need to be positive or both need to be negative.

You can have `a times b = 10` for e.g.
`a = 10" "` and `" "b = 1" "` (or the negative values)
`a = 5" "` and `" "b = 2" "` (or the negative values)

Checking the sum you can see that it works for `a = -5` and `b = -2`:

`a + b = (-5) + (-2) = -7`

`a times b = (-5) times (-2) = 10`

Thus, your factorization is:

`y = x^2 - 7x + 10 = (x - 5)(x - 2)`

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By the way, this also means that the equation

`x^2 - 7x + 10 = 0`

`<=> (x-5)(x-2) = 0`

has two solutions:

`x = 5 " or "x = 2`