Question

Given: p-q=6, pq=36
To find: p+q

Answer 1

GIVEN :–

• p - q = 6

• pq = 36

TO FIND :–

• Value of p + q = ?

SOLUTION :–

$\\ \sf \implies \: p - q = 6$

• Square on both sides –

$\\ \sf \implies \: (p - q) ^{2} = {(6)}^{2}$

$\\ \sf \implies \: {p}^{2} + {q}^{2} - 2pq = 36 \: \: \: \: [ \: \because \: {(a - b)}^{2} = {a}^{2} + {b}^{2} - 2ab ]$

• Put the value –

$\\ \sf \implies \: {p}^{2} + {q}^{2} - 2(36) = 36$

$\\ \sf \implies \: {p}^{2} + {q}^{2} - 72 = 36$

$\\ \sf \implies \: {p}^{2} + {q}^{2} = 36 + 72$

$\\ \sf \implies \: {p}^{2} + {q}^{2} = 108 \: \: \: \: \: - - - eq.(1)$

• Now let's find the value of (p + q) –

$\\ \sf \implies \: (p + q)^{2} = {p}^{2} + {q}^{2} + 2pq$

• We should write this as –

$\\ \sf \implies \: p + q = \sqrt{ {p}^{2} + {q}^{2} + 2pq}$

• Using eq.(1) –

$\\ \sf \implies \: p + q = \sqrt{ 108 + 2(36)}$

$\\ \sf \implies \: p + q = \sqrt{ 108 + 72}$

$\\ \sf \implies \: p + q = \sqrt{ 180}$

$\\ \sf \implies \: p + q = \sqrt{5 \times 36}$

$\\ \implies \large { \boxed{ \sf p + q =6 \sqrt{5 } }}$

• Hence , The value of (p + q) = $\sf = 6 \sqrt{5}$

Answer 2

Given:

• p - q = 6
• pq = 36

To find:

• p + q

Solution:

Taking the given equations

p - q = 6 ……… eq(i)

pq = 36 ……… eq(ii)

Taking equation 1 and squaring on both sides

→ (p - q)² = 6²

→ p² - q² - 2pq = 36

Already given that pq = 36

→ p² - q² - 2(36) = 36

→ p² - q² - 72 = 36

→ p² - q² = 108 ……… eq ( iii )

Now, let

p + q be x

p + q = x

Squaring on both sides

→ (p + q)² = x²

→ p² + q² + 2pq = x²

→ 108 + 2(36) = x²

→ 180 = x²

→ x = √180

→ x = 6√5

Hence , p + q = 6√5