Create a formula by given solutions

Question

For my upcoming middle school exams I will need to convert a formula.

I have got the following question:

Create a formula which has the following solutions: $$ x_{1} = 5,\quad x_{2} = -3.$$

The solutions say that $$ (x−5)⋅(x+3)=0 $$ is possible. But they don't explain how they did it.

My question is also: How can I create that formula by myself?

Answer 1

First, can you understand that $x_1 = 5$, and $x_2 = -3$ are solutions to

$$(x - 5)(x + 3)= 0\quad ?$$

Just substitute $x = 5$ and see that you get, as desired $0 = 0$.

Similarly, substitute $x = -3$, then $(x - 5)(x + 3) = (-3 - 5)(-3 + 3) = -8 \times 0 = 0\quad \checkmark$

Suppose you are given two values which are said to be solutions to some equation.

Say $\,x_1 = a\,$ and $\,x_2 = b\;$ are the given solutions; then we will always have that $$(x - a)(x - b) = 0$$

So in the case of $\;x_1 = 5,\quad x_2 = -3$, we have $$(x - 5)(x - (-3)) = 0 $$ $$\iff (x - 5)(x+3) = 0$$

This is true in general: If your are given that $x_1, \;x_2,\; ... x_n$ are solutions to some equation, then it will always be the case that $$\underbrace{(x - x_1)(x-x_2)\cdots (x - x_n)}_{n\; \text{ factors}} = 0$$

Answer 2

If your solutions are $x_1,...x_n$

An easy way is $(x-x_1)...(x-x_n)$

Here your $x_1=5,x_2=-3$ using the formula gives $(x-5)(x--3)=(x-5)(x+3)$

Of course, I am, in no way, suggesting this is the only way to create an equation to which has only these two numbers as solutions, but it is the simplest polynomial.