Two numbers have a sum of 22 and their product is 103. What are the numbers in simplest radical form?

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Two numbers have a sum of 22 and their product is 103. What are the numbers in simplest radical form?

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Answer 1 · 2018-06-19T15:04:39

$11 \pm 3 \sqrt{2}$ $11+-3sqrt(2)$

Explanation:

Call the two numbers $a$ $a$ and $b$ $b$ . Then:
$a + b = 22$ $a+b=22$ and $a b = 103$ $ab=103$ .

From the first equation:
$b = 22 - a$ $b=22-a$
Substitute into the second:
$a (22 - a) = 103$ $a(22-a)=103$
$22 a - a^{2} = 103$ $22a-a^2=103$
$0 = a^{2} - 22 a + 103$ $0=a^2-22a+103$

Quadratic formula:
$a = \frac{1}{2} (22 \pm \sqrt{484 - 412}) = \frac{1}{2} (22 \pm \sqrt{72})$ $a=1/2(22+-sqrt(484-412))=1/2(22+-sqrt(72))$
$= \frac{1}{2} (22 \pm 6 \sqrt{2}) = 11 \pm 3 \sqrt{2}$ $=1/2(22+-6sqrt(2))=11+-3sqrt(2)$

So we have two solutions for $a$ $a$ . Use these to obtain $b$ $b$ from the first formula:
$11 \pm 3 \sqrt{2} + b = 22$ $11+-3sqrt(2)+b=22$
$b = 11 ¯ ¯¯¯ ¯ + 3 \sqrt{2}$ $b=11bar(+)3sqrt(2)$

So for the pair of values of $a$ $a$ , $b$ $b$ always takes the other one of the two values. So the two numbers are simply the pair:
$11 \pm 3 \sqrt{2}$ $11+-3sqrt(2)$

Double check with the second formula that these are correct:
$(11 + 3 \sqrt{2}) (11 - 3 \sqrt{2}) = 103$ $(11+3sqrt(2))(11-3sqrt(2))=103$
Note that this is a 'difference of two squares' formula
$121 - 9 \cdot 2 = 103$ $121-9*2=103$
Yep, this checks out.

Answer 2 · 2018-06-19T15:07:48

$a = 11 + 3 \sqrt{2}$ $a=11+3sqrt2$
$b = 11 - 3 \sqrt{2}$ $b=11-3sqrt2$

Explanation:

$a + b = 22 \Rightarrow b = 22 - a$ $a+b=22 => b=22-a$
$a b = 103$ $ab=103$
$a \cdot (22 - a) = 103$ $a*(22-a)=103$
$- a^{2} + 22 a = 103$ $-a^2+22a=103$
$a^{2} - 22 a + 103 = 0$ $a^2-22a+103=0$
We have ${(a - 11)}^{2} - 121 + 103 = {(a - 11)}^{2} - 18$ $(a-11)^2-121+103=(a-11)^2-18$
$a - 11 = \pm \sqrt{18} = \pm 3 \sqrt{2}$ $a-11=+-sqrt18=+-3sqrt2$
$a = 11 + 3 \sqrt{2}$ $a=11+3sqrt2$
$b = 11 - 3 \sqrt{2}$ $b=11-3sqrt2$

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Two numbers have a sum of 22 and their product is 103. What are the numbers in simplest radical form?

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