Answer:
the value of pq is 1/2.
Explanation:
To find the value of p² + q² and pq, we can use the given equations:
1. p + q = √17
2. p - q = √15
To solve this system of equations, we can use the method of substitution. Let's solve equation 2 for p:
p = √15 + q
Now substitute this value of p into equation 1:
(√15 + q) + q = √17
Combine like terms:
2q + √15 = √17
Now, isolate q by subtracting √15 from both sides:
2q = √17 - √15
Next, divide both sides by 2 to solve for q:
q = (√17 - √15) / 2
Now that we have the value of q, we can substitute it back into equation 1 to solve for p:
p + (√17 - √15) / 2 = √17
Multiply both sides by 2 to eliminate the fraction:
2p + √17 - √15 = 2√17
Subtract √17 from both sides:
2p - √15 = 2√17 - √17
Now, isolate p by subtracting √15 from both sides:
2p = 2√17 - √17 + √15
Simplify:
2p = √17(2 - 1) + √15
2p = √17 + √15
Divide both sides by 2 to solve for p:
p = (√17 + √15) / 2
Now that we have the values of p and q, we can calculate p² + q² and pq:
p² + q² = (√17 + √15)² + (√15 - √17)²
Simplify the expressions:
p² + q² = 17 + 2√15√17 + 15 + 15 - 2√15√17 + 17
Combine like terms:
p² + q² = 64
So, p² + q² is equal to 64. To find the value of pq, we can multiply the values of p and q:
pq = (√17 + √15) / 2 * (√17 - √15) / 2 Simplify the expression:
pq = (17 - 15) / 4 pq = 2 / 4 pq = 1/2 Therefore, the value of pq is 1/2.