Question

the sum of two numbers are 8 and their product is 12

Answer 1

Answer:

Begin by creating the two equations that are provided by the information in the question.

x

+

y

=

8

and

x

y

=

15

Now you can solve logically because of the simplicity of the numbers, by listing the factors of 15 and determine which pair will add up to 8

1 and 15

3 and 5

The Answer is 3 and 5

Or you can solve algebraically by substitution

x

+

y

=

8

can be converted to

y

=

8

−

x

Now substitute the first equation into the second equation for the y variable.

x

(

8

−

x

)

=

15

Distribute

8

x

−

x

2

=

15

Set the equation equal to zero.

8

x

−

x

2

−

15

=

15

−

15

Rearrange to put the squared term first.

−

x

2

+

8

x

−

15

=

0

Factor the trinomial.

-1(x-5)(x-3) = 0#

Set both factors equal to zero.

x

−

5

=

0

x

=

5

x

−

3

=

0

x

=

3

5

and

3

Answer 2

Given,

The sum of two numbers$\[ = 8\]$

The product of two numbers$\[ = 12\]$

Solution,

Assume that the numbers are x and y respectively.

Therefore,

$\[\begin{array}{l} \Rightarrow x + y = 8\\ \Rightarrow xy = 12\end{array}\]$

So, the value of x is

$\[\begin{array}{l} \Rightarrow x + \frac{{12}}{x} = 8\\ \Rightarrow {x^2} + 12 = 8x\\ \Rightarrow {x^2} - 8x + 12 = 0\\ \Rightarrow {x^2} - \left( {6 + 2} \right)x + 12 = 0\end{array}\]$

Solve further,

$\[\begin{array}{l} \Rightarrow {x^2} - 6x - 2x + 12 = 0\\ \Rightarrow x\left( {x - 6} \right) - 2\left( {x - 6} \right) = 0\\ \Rightarrow \left( {x - 2} \right)\left( {x - 6} \right) = 0\\ \Rightarrow x = 2,6\end{array}\]$

If we consider $\[x = 2\]$ then $\[y = 6\]$ and vise -versa.

Hence, the numbers are $\[2\,{\rm{and}}\,6.\]$