Difference between revisions of "Sub-Problem 2"

Latest revision as of 21:23, 28 November 2023

(b) Determine all $(a,b)$ such that: $\sqrt{a} + \sqrt{b} = 8$ $\log_{10} a + \log_{10} (b) = 2$

From equation 2, we can acquire ab = 100

We can then expand both sides by squaring:

$(\sqrt{a} + \sqrt{b})^2 = (8)^2$ $(a + b + 2 \sqrt{ab}) = 64$

since ab = 100: 2root(ab) is 2root(100), which is 20.

We can get the below equation:

$(a + b) = 44$ $(ab) = 100$

Substitue b = 44 - a, we get

$((44-a)a) = 100$ $(44a - a^2 - 100) = 0$

By quadratic equations Formula:

${a=\frac{-44 \pm \sqrt{44^2-4(-1)(-100)}}{2(-1)}}$

${a=\frac{44 \pm \sqrt{1536}}{2(1)}}$

which leads to the answer of 22 +- 8\sqrt(6)

Since a = 44 - b, two solutions are:

$(a,b) = (22 + 8\sqrt6, 22 - 8\sqrt6)$ $(a,b) = (22 - 8\sqrt6, 22 + 8\sqrt6)$

~North America Math Contest Go Go Go

~North America Math Contest Go Go Go

@@ Line 1: / Line 1: @@
 == Problem ==
-(b) Determine all (a,b) such that:
+(b) Determine all <math>(a,b)</math> such that:
-                   <cmath>\sqrt{a} + \sqrt{b} = 8</cmath> <cmath>\log_{10} a +  \log_{10} (b) = 2</cmath>
+<cmath>\sqrt{a} + \sqrt{b} = 8</cmath> <cmath>\log_{10} a +  \log_{10} (b) = 2</cmath>
-[https://artofproblemsolving.com/wiki/index.php/Problem_7#Problem_7 Go back to the previous page]
+== Solution 1 ==
+From equation 2, we can acquire ab = 100
+We can then expand both sides by squaring:
+<cmath>(\sqrt{a} + \sqrt{b})^2 = (8)^2</cmath>
+<cmath>(a + b + 2 \sqrt{ab}) = 64</cmath>
+since ab = 100: 2root(ab) is 2root(100), which is 20.
+We can get the below equation:
+<cmath>(a + b) = 44</cmath>
+<cmath>(ab) = 100</cmath>
+Substitue b = 44 - a, we get
+<cmath>((44-a)a) = 100</cmath>
+<cmath>(44a - a^2 - 100) = 0</cmath>
+By quadratic equations Formula:
+<math>{a=\frac{-44 \pm \sqrt{44^2-4(-1)(-100)}}{2(-1)}}</math>
+<math>{a=\frac{44 \pm \sqrt{1536}}{2(1)}}</math>
+which leads to the answer of 22 +- 8\sqrt(6)
+Since a = 44 - b, two solutions are:
+<cmath>(a,b) = (22 + 8\sqrt6, 22 - 8\sqrt6)</cmath>
+<cmath>(a,b) = (22 - 8\sqrt6, 22 + 8\sqrt6)</cmath>
-== Solution 1 ==
+~North America Math Contest Go Go Go
 == Video Solution ==
+https://www.youtube.com/watch?v=C180TL1PLaA
+~North America Math Contest Go Go Go