Dark clouds approached the adventure training camp as Emek and Tara raced to unlock the communications station. Its three-part access code came from different equations. The first screen showed x^2 - 11x + 28 = 0. Tara factored it as (x - 4)(x - 7) = 0, and a note requiring x greater than 5 left 7 as the first code number. The second screen displayed (y + 3)/(y - 2) = 4. Emek cleared the denominator and found y = 11/3, checking that y was not the forbidden value 2. The final screen showed 3^(z - 1) = 27. Recognizing 27 as 3^3, Tara immediately found z = 4. The station instructed them to multiply the three valid results. Their calculation gave 7 times 11/3 times 4, or 308/3. They entered the exact value instead of rounding. The communications system activated just as rain began, allowing the team to receive the storm safety instructions.

Challenges

Start
Challenge 1 of 5

Challenges

Start
Challenge 1 of 4
Hint
Separate the solutions of the equation from the additional restriction on x.
A student solves x^2 - 11x + 28 = 0 and says both 4 and 7 can be used for the first code number. Which response correctly identifies the mistake?
Only 4 solves the quadratic equation.Neither value solves the quadratic equation.Both values solve the equation, but the condition x greater than 5 eliminates 4, leaving 7.The larger solution must always be rejected in a quadratic equation.
Correct. Both 4 and 7 solve the quadratic, but only 7 satisfies x greater than 5.
Check both the factored equation and the stated restriction before choosing the code value.
Next
🎉 Congrats! You got this question!
Hint
Rewrite 27 as a power of 3 and compare exponents.
For the final screen, 3^(z - 1) = 27. What integer value of z makes the equation true?
Correct. Since 27 = 3^3, z - 1 = 3 and z = 4.
Express 27 using base 3, then set the exponents equal.
Next
🎉 Congrats! You got this question!
Hint
Multiply both sides by y - 2, solve the resulting linear equation, and then check that the denominator is not zero.
Emek wants to verify the second code value before multiplying the three results. Which calculation correctly solves (y + 3)/(y - 2) = 4 and checks the restriction?
y + 3 = 4y - 8, so 11 = 3y and y = 11/3; since 11/3 is not 2, it is valid.y + 3 = 4y - 2, so 5 = 3y and y = 5/3; since 5/3 is not 2, it is valid.y + 3 = 4(y - 2), so y = 2; because 2 solves the equation, it is valid.y + 3 = 4y + 8, so y = -5/3; since the denominator is then zero, it is invalid.
Correct. The solution is 11/3, and it does not violate the restriction y not equal to 2.
Clear the denominator carefully, then verify that the resulting value does not make y - 2 equal to zero.
Next
🎉🎉🎉🎉🎉🎉🎉🎉
🎉 Congrats! You finished this challenge set!
Son Keşif Kodu
TR
ARC_G12_OPS_02_STORY_03
Son Keşif Kodu
The Final Expedition Code
EN
ARC_G12_OPS_02_STORY_03
The Final Expedition Code