Question

if X+Y=5 XY find X-Y using identities

Answer 1

$\huge \mathbb{ \pink{Q} \red{U}\pink{E}\red{S}\pink{T}\red{I}\pink{O}\red{N} : }$

• If x + y = 5, xy = 4, find x - y (using identities).

$\huge \text{A\green{N}S\green{W}E\green{R} :}$

Here,

x + y = 5

• Squaring both sides,

→ (x + y)² = 5²

→ x² + y² + 2xy = 25

→ x² + y² + 2(4) = 25

→ x² + y² + 8 = 25

→ x² + y² = 25 + 8

→ x² + y² = 33

Now,

$\rm{}x - y \: \: \: = {(x - y)}^{2} \\ \\ \sf = > {x}^{2} + {y}^{2} - 2xy$

• Putting the values in above expression, we get,

$\large \bf = > 33 + 2(4) \\ \large \bf = > \: 33 + 16 \\ \large \bf = > \: \: \red{49}$

$<bold>___________________________</bold>$

♦ Some Identities --

• (x + y)² = x² + y² + 2xy
• (x - y)² = x² + y² - 2xy
• x² - y² = (x + y)(x - y)
• x² + y² = (x + y)² -2xy
• (x + a)(x + b) = x² +(a + b)x + ab
• (x + y)³ = x³ + y³ + 3xy(x + y)
• (x - y)³ = x³ - y³ - 3xy(x - y)
• x³ - y³ = (x - y)(x² + y² + xy)
• x³ + y³ = (x + y)(x² + y² - xy)