Math Is Fun Forum

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Square Root Method

Solve ( x − 2)^2 = 49 using the Square Root Method.

Let me see.

I know that taking the square root on both sides of the equation is step 1.

sqrt{( x − 2)^2} = sqrt{49}

x - 2 = -7 or x - 2 = 7

x = -7 + 2 or x = 7 + 2

x = -5 or x = 9

Question:

Why do we get a negative answer and a positive answer when taking the square root?

Questions:

Do I use or? Do I use and?

Take the answers here.

Do I write the answers as
x = -5 or x = 9?

Do I write the answer as
x = -5 and x = 9?

You say?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Square Root Method

Every positive number (except 0) has two square roots, one positive and one negative, so you need to allow for both possibilities.  Sometimes the context of the question will then rule out one answer.

x cannot be both -5 and 9 at the same time so it has to be or

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Re: Square Root Method

Every positive number (except 0) has two square roots, one positive and one negative, so you need to allow for both possibilities.  Sometimes the context of the question will then rule out one answer.

x cannot be both -5 and 9 at the same time so it has to be or

Bob

Why does every positive number have two square roots?

Take 7, for example. Are you saying that 7 has both -sqrt{7} and sqrt{7}?

Why is this the case?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Square Root Method

Is a number (A) the square root  of another number (B) ?

The only requirement is that A^2 = B.

As -√ 7 x -√ 7 = 7 it is a square root of 7.

If √r is a square root of y, then -√r times  -√r = y as multiplying two minuses makes a plus.

So this happens for every positive number.

Negative numbers don't have a real number root.  (But see complex numbers for more on this)

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Re: Square Root Method

Is a number (A) the square root  of another number (B) ?

The only requirement is that A^2 = B.

As -√ 7 x -√ 7 = 7 it is a square root of 7.

If √r is a square root of y, then -√r times  -√r = y as multiplying two minuses makes a plus.

So this happens for every positive number.

Negative numbers don't have a real number root.  (But see complex numbers for more on this)

Bob

In your reply, where did sqrt{r} come from?
Where did the variable [ r ]come from?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Square Root Method

This often happens in maths. You suspect a general result is true. Step 1**: Try out some particular values to see if it's true in those cases. Step 2: Then try to show it's always true by substituting letters for numbers.

I've shown that it's true for root 7.  Now to test in general. So choose a letter to replace 7.  At first I chose x but then I got into trouble using x to mean times as well. So I switched to a different letter. I chose 'r' to stand for 'root'.

Bob

** Strictly if you can do step 2 you don't need step 1, as step 2 shows it's always true.  But, if you're struggling to understand a topic, trying number values first helps.

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Re: Square Root Method

This often happens in maths. You suspect a general result is true. Step 1**: Try out some particular values to see if it's true in those cases. Step 2: Then try to show it's always true by substituting letters for numbers.

I've shown that it's true for root 7.  Now to test in general. So choose a letter to replace 7.  At first I chose x but then I got into trouble using x to mean times as well. So I switched to a different letter. I chose 'r' to stand for 'root'.

Bob

** Strictly if you can do step 2 you don't need step 1, as step 2 shows it's always true.  But, if you're struggling to understand a topic, trying number values first helps.

Thanks for clearing that up.