Question

find the solutions to x²=12

Answer 1

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x2-12=0 Two solutions were found :                   x = 2 • ± √3 = ± 3.4641

Step by step solution :Step  1  :Trying to factor as a Difference of Squares :

 1.1      Factoring:  x2-12 

Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =

         A2 - AB + BA - B2 =

         A2 - AB + AB - B2 = 

         A2 - B2

Note :  AB = BA is the commutative property of multiplication. 

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 12 is not a square !! 

Ruling : Binomial can not be factored as the difference of two perfect squares.

Equation at the end of step  1  : x2 - 12 = 0 Step  2  :Solving a Single Variable Equation :

 2.1      Solve  :    x2-12 = 0 

 Add  12  to both sides of the equation : 

                      x2 = 12 

 

 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  

                      x  =  ± √ 12  

 Can  √ 12 be simplified ?

Yes!   The prime factorization of  12   is

   2•2•3  

To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 12   =  √ 2•2•3   =

                ±  2 • √ 3 

The equation has two real solutions  

 These solutions are  x = 2 • ± √3 = ± 3.4641  

 

Two solutions were found :                   x = 2 • ± √3 = ± 3.4641

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Answer 2

${x}^{2} = 12$

As we know,

3^2<12<4^2

9<12<16

So we can say that answer is between 3^2 and 4^2

3^2=9

3.1^2=9.61

3.2^2=10.24

3.5^2=12.25

Here 3.5^2= 12.25>12

Let's take a smaller unit=3.4^2

3.4^2=11.56

Here

3.4^2<12<3.5^2

As we see that 3.5^2 is much nearer to 12 hence 3.5 is the answer.

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