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yKpoI14uk [10]
3 years ago
15

How to find a rule and extend a pattern

Mathematics
1 answer:
makvit [3.9K]3 years ago
7 0
    Look for a relationship between the input “X” side, and the output “Y” side. Check thepattern<span> against every row. It has to be true for the whole table, or it's not the </span>rule<span>. Use the </span>rule<span>, and the numbers you </span>know<span>, to complete or </span>extend<span> the </span>pattern<span>.</span>
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Create a list of steps, in order, that will solve the following equation 1/4(x+5)^2-1=3
julia-pushkina [17]

Answer:

either x = -1, or x = - 9

Step-by-step explanation:

given   \frac{1}{4} (x+5)^{2} -1=3

add one to both sides   \frac{1}{4} (x+5)^{2} =4

im not sure which one comes first. either you multiply both the x and the 5 in the parenthesis by \frac{1}{4}. or you square x and 5

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :   1/4*(x+5)^2-(4)=0  

Step by step solution :

Step  1  :

           1 /4

Equation at the end of step  1  :

  1                  

 (— • (x + 5)2) -  4  = 0  

  4                  

Step  2  :

Equation at the end of step  2  :

 (x + 5)2    

 ———————— -  4  = 0  

    4        

Step  3  :

Rewriting the whole as an Equivalent Fraction :

3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  4  as the denominator :

        4         4 • 4

   4 =  —  =  ———

        1            4  

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

3.2       Adding up the two equivalent fractions

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

(x+5)2 - (4 • 4)     x2 + 10x + 9

———————     ———————  =  ——

       4                        4      

Trying to factor by splitting the middle term

3.3     Factoring  x2 + 10x + 9  

The first term is,  x2  its coefficient is  1 .

The middle term is,  +10x  its coefficient is  10 .

The last term, "the constant", is  +9  

Step-1 : Multiply the coefficient of the first term by the constant   1 • 9 = 9  

Step-2 : Find two factors of  9  whose sum equals the coefficient of the middle term, which is   10 .

     -9    +    -1    =    -10  

     -3    +    -3    =    -6  

     -1    +    -9    =    -10  

     1    +    9    =    10    That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  1  and  9  

                    x2 + 1x + 9x + 9

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

                   x • (x+1)

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

                   9 • (x+1)

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

                   (x+9)  •  (x+1)

            Which is the desired factorization

Equation at the end of step  3  :

 (x + 9) • (x + 1)

 —————————————————  = 0  

         4        

Step  4  :

When a fraction equals zero :

4.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

 (x+9)•(x+1)

 ——————————— • 4 = 0 • 4

      4      

Now, on the left hand side, the  4  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :

  (x+9)  •  (x+1)  = 0

Theory - Roots of a product :

4.2    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 :

4.3      Solve  :    x+9 = 0  

Subtract  9  from both sides of the equation :  

                     x = -9

Solving a Single Variable Equation :

4.4      Solve  :    x+1 = 0  

Subtract  1  from both sides of the equation :  

                     x = -1

Supplement : Solving Quadratic Equation Directly

Solving    x2+10x+9  = 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

6 0
3 years ago
Write an equation for the description.<br> The difference between a number x and 19 is 27.
ra1l [238]

Answer

x-19=27

Step-by-step explanation:

4 0
3 years ago
The daily demand for ice cream cones at a price of $1.20 per cone is 50 cones. At a price of $2.20 per cone, the demand is 30 co
Lapatulllka [165]

Answer:

<h2>The demand at a price of $1.50 per cone is 44 cones</h2>

Step-by-step explanation:

 let us apply the interpolation formula to solve this problem

we have

\frac{y-y1}{x-x1} =\frac{y2-y1}{x2-x1}

Now let us set the values to the notation above

x= $1.20

x1= $2.20

x2= $1.50

y = 50 cones

y1= 30 cones

y2= ?

y2= \frac{(y-y1)}{(x-x1)} (x2-x1)+y1

Substituting our given data we have

y2= \frac{(50-30)}{(1.2-2.2)} (1.5-2.2)+30\\\\y2=  \frac{20}{-1} (-0.7)+30\\\\y2=   (20*0.7)+30\\y2= 14+30\\y2= 44

the demand at a price of $1.50 per cone is 44 cones

4 0
3 years ago
M^2-10m+9=(m-9)( ) filling the missing factor to complete the equation
Romashka [77]

Answer:

m-1

Step-by-step explanation:

3 0
3 years ago
Frank devised a formula for his catering business that calculates the number of meatballs he needs to prepare. The formula is m=
NeX [460]
So if m=4a+2c is your equation then all you have to do is plug in the numbers.

So you then get m=4(25)+2(7), for 25 adults and 7 children.

Then you solve.  m=100+2(7)
                           m=100+14
                            m=114
So Frank will need 114 Meatballs for the party of 25 adults and 7 children
 Hope this helps! And if you need anything else just ask :)


5 0
3 years ago
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