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Sever21 [200]
3 years ago
14

Tom made 5,000 mL of soup. He packed 2.5 liters of the soup in his kids' lunches. He froze the rest of the soup. How many millil

iters of soup did Tom have left to freeze?
Mathematics
1 answer:
MakcuM [25]3 years ago
7 0

Answer:

Amount of frozen soup = 2500 mL

Step-by-step explanation:

We are told that tom made 5,000 mL of soup.

We are also told he packed 2.5 Litres into his kids' lunches.

Now, from conversions,

1 liter = 1000 mL

Thus,

2.5 litres = 2.5 × 1000 = 2500 mL

Now, we are told he froze the rest of the soup.

Thus;

Amount of frozen soup = total amount of soup made - amount given to kids' lunch pack.

Amount of frozen soup =

5000 - 2500 = 2500 ml

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What are the solutions of the equation x4-9x^2+8=0 use
inysia [295]

Answer:

x = 2 • ± √2 = ± 2.8284

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Step-by-step explanation:

x4-9x2+8=0  

Four solutions were found :

x = 2 • ± √2 = ± 2.8284

x = 1

x = -1

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "x2"   was replaced by   "x^2".  1 more similar replacement(s).

Step by step solution :

Step  1  :

Equation at the end of step  1  :

 ((x4) -  32x2) +  8  = 0  

Step  2  :

Trying to factor by splitting the middle term

2.1     Factoring  x4-9x2+8  

The first term is,  x4  its coefficient is  1 .

The middle term is,  -9x2  its coefficient is  -9 .

The last term, "the constant", is  +8  

Step-1 : Multiply the coefficient of the first term by the constant   1 • 8 = 8  

Step-2 : Find two factors of  8  whose sum equals the coefficient of the middle term, which is   -9 .

     -8    +    -1    =    -9    That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -8  and  -1  

                    x4 - 8x2 - 1x2 - 8

Step-4 : Add up the first 2 terms, pulling out like factors :

                   x2 • (x2-8)

             Add up the last 2 terms, pulling out common factors :

                    1 • (x2-8)

Step-5 : Add up the four terms of step 4 :

                   (x2-1)  •  (x2-8)

            Which is the desired factorization

Trying to factor as a Difference of Squares :

2.2      Factoring:  x2-1  

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 : 1 is the square of 1

Check :  x2  is the square of  x1  

Factorization is :       (x + 1)  •  (x - 1)  

Trying to factor as a Difference of Squares :

2.3      Factoring:  x2 - 8  

Check : 8 is not a square !!  

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

Equation at the end of step  2  :

 (x + 1) • (x - 1) • (x2 - 8)  = 0  

Step  3  :

Theory - Roots of a product :

3.1    A product of several terms equals zero.  

When a product of two or more terms equals zero, then at least one of the terms must be zero.  

We shall now solve each term = 0 separately  

In other words, we are going to solve as many equations as there are terms in the product  

Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

3.2      Solve  :    x+1 = 0  

Subtract  1  from both sides of the equation :  

                     x = -1  

Solving a Single Variable Equation :

3.3      Solve  :    x-1 = 0  

Add  1  to both sides of the equation :  

                     x = 1  

Solving a Single Variable Equation :

3.4      Solve  :    x2-8 = 0  

Add  8  to both sides of the equation :  

                     x2 = 8  

 

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

                     x  =  ± √ 8  

Can  √ 8 be simplified ?

Yes!   The prime factorization of  8   is

  2•2•2  

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).

√ 8   =  √ 2•2•2   =

               ±  2 • √ 2  

The equation has two real solutions  

These solutions are  x = 2 • ± √2 = ± 2.8284  

 

Supplement : Solving Quadratic Equation Directly

Solving    x4-9x2+8  = 0   directly  

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Solving a Single Variable Equation :

Equations which are reducible to quadratic :

4.1     Solve   x4-9x2+8 = 0

This equation is reducible to quadratic. What this means is that using a new variable, we can rewrite this equation as a quadratic equation Using  w , such that  w = x2  transforms the equation into :

w2-9w+8 = 0

Solving this new equation using the quadratic formula we get two real solutions :

  8.0000  or   1.0000

Now that we know the value(s) of  w , we can calculate  x  since  x  is  √ w  

Doing just this we discover that the solutions of  

  x4-9x2+8 = 0

 are either :  

 x =√ 8.000 = 2.82843  or :

 x =√ 8.000 = -2.82843  or :

 x =√ 1.000 = 1.00000  or :

 x =√ 1.000 = -1.00000  

Four solutions were found :

x = 2 • ± √2 = ± 2.8284

x = 1

x = -1

4 0
3 years ago
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