Question

Two numbers differ by 9. The sum of their squares is 653. Find the numbers.

Answer 1

Let the two numbers be a and b
\[a-b=9 \]\[a^2+b^2=653 \]

Answer 2

And I should rearrange the a and the b to plug it into the second equation, right?

Answer 3

yes

Answer 4

How do you factor 2b^2+188b-653?

Answer 5

first we need to know factors of 653
... can't be 3 or 11.

.....try 7

Answer 6

7 doesn't work

Answer 7

13

Answer 8

***how about 13

you could always use ...
\[\Large x= {-b \pm \sqrt{b^2-4ac} \over 2a}\]

Answer 9

Sorry, it's been about 2 years since I've used that formula. Can you just remind me how to do it? @PaxPolaris

Answer 10

sorry.... didn't notice that the quadratic equation you have is actually Wrong.

\[\large a^2+b^2=653\]\[\implies \large \left( b+9 \right)^2+b^2=653\ \dots\ \dots \left\{ \because a=b+9 \right\}\]\[\implies \Large \left( b^2+18b+81 \right)+b^2=653\]implies Large \left( b^2+18b+81 \right)+b^2=653

Answer 11

\[\implies \Large 2b^2+18b+81 -653=0\]\[\implies \LARGE 2b^2+18b-572=0\]

.... then finally you can divide everything by 2:\[\implies b^2+9b-286=0\]