Question 916182
A positive number is divided into two parts such that the sum of the squares of the two parts is 20. The square of the larger part is 8 times the smaller part. Taking x as the smaller part of the two parts, find the number.
Please help me to solve this problem which is based on the quadratic equation.
<pre>
n = the number
x = smaller part
n-x = larger part

</pre>
the sum of the squares of the two parts is 20
<pre>
{{{x^2+(n-x)^2=20}}}
</pre>
The square of the larger part is 8 times the smaller part
<pre>
{{{(n-x)^2=8x}}}

So we have this system of equations to solve:

{{{system(x^2+(n-x)^2=20,(n-x)^2=8x)}}}

Using the second, substitute 8x for (n-x)<sup>2</sup> in the first

{{{x^2+8x=20}}}

Get 0 on the right by subtracting 20 from both sides:

{{{x^2+8x-20=0}}}

Factor:

{{{(x-2)(x+10)=0}}}

Use the zero-factor property by setting each factor = 0

x-2 = 0;   x+10 = 0
  x = 2       x = -10

We ignore the negative answer. 

x = smaller part = 2
</pre>
 find the number.
<pre>
Substitute x = 2 in

{{{(n-x)^2=8x}}}

{{{(n-2)^2=8(2)}}}

{{{(n-2)^2=16}}}

Use the principle of square roots:

{{{n-2= ""+-sqrt(16)}}}

{{{n-2 = "" +- 4}}}

Add 2 to both sides

{{{n= "2" +- 4}}}

Using the +, we get 2+4 = 6
Using the -, we get 2-4 = -2

We ignore the negative answer.

Solution: 6

Checking:

The two parts of 6 are 2 and 4
</pre>
the sum of the squares of the two parts is 20.
<pre>
4<sup>2</sup>+2<sup>2</sup> = 16 + 4 = 20

That checks.
</pre>
The square of the larger part is 8 times the smaller part
<pre>
4<sup>2</sup> = 16 and 16 = (8)(2)

So that checks.

Edwin</pre>