Question

Find the value of p²+q² and pq when p+q =√17 and p-q = √15

Answer 1

`p+q=√(17)\\\mathrm{Squaring\ both\ sides,}\\(p+q)^2=17\\\mathrm{or,\ }p^2+2pq+q^2=17........(1)\\\mathrm{Also\ we\ have,}\\p-q=√(15)\\\mathrm{Squaring\ both\ sides, }\\(p-q)^2=15\\\mathrm{or,\ }p^2-2pq+q^2=15......(2)\\\mathrm{Adding\ equations(1)\ and\ (2),}\\2p^2+2q^2=15+17\ \mathrm{or,\ }2(p^2+q^2)=32\\\mathrm{or,\ }p^2+q^2=16`

`\mathrm{We\ know,}\\p^2+q^2=(p+q)^2-2pq\\\mathrm{or,\ }16=17-2pq\\\mathrm{or,\ }2pq=1\\\mathrm{or,\ }pq=0.5`

Answer 2

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.