An algebra problem by Mohammad Khaza

given that .

x-y=2 and xy=63

so, x^2+y^2

 =(x-y)^2+2 .x . y

=4+126

=130

Solution 1

Given that ,

$•$ $x - y = 2$

$•$ $x y = 63$

Now,

$x^2 + y^2$

$= (x - y)^2 + 2 x y$

$= (2)^2 + 2(63)$

$\Rightarrow 4 + 126 = \boxed {130}$

Solution 2

Working on $(x - y) = 2$

$(x - y) = 2$

$\Rightarrow (x - y)^2 = 2^2$

Applying the formula $(a - b)^2 = a^2 - 2ab + b^2$

$\Rightarrow (x - y)^2 = x^2 - 2xy + y^2 = 4$

It is given that $xy = 63$

$\Rightarrow x^2 + y^2 = 4 + 2xy$

$\Rightarrow x^2 + y^2 = 4 + 2 \times 63 \implies 4 + 126 = \boxed{130}$

$\begin{aligned} x - y & = 2 & \small \color{#3D99F6} \text{Given} \\ (x - y)^2 & = 2^2 \\ x^2 - 2xy + y^2 & = 4 \\ x^2 + y^2 & = 4 + 2\color{#3D99F6}xy & \small \color{#3D99F6} \text{Note that }xy = 63 \\ & = 4 + 2\color{#3D99F6}(63) \\ & = \boxed{130} \end{aligned}$

Given that: x-y=2

xy=63

x^2+y^2=? ..............................................................................

Let,

x-y=2 ... (1)

xy=63 ... (2)

From (1), make x as the subject.

x=y+2 ... (3)

Substitute (3) into (2),

(y+2)y=63

y^2+2y=63

y^2+2y-63=0

Factorise this equation,

(y+9)(y-7)=0

y=-9,7

When y=-9, (substitute this into (1))

x-(-9)=2

x=-7

[LET THIS TO BE SOLUTION 1]

When y=7, (substitute this into (1))

x-(7)=2

x=9

[LET THIS TO BE SOLUTION 2]

Substitute solution 1 into x^2+y^2=?

(-7)^2+(-9)^2=?

?=49+81

?=130#

..................................OR.................................... Substitute solution 2 into x^2+y^2=?

(9)^2+(7)^2=?

?=81+49

?=130#

$x - y = 2$ and $xy = 63$

$(x - y)^2 = x^2 + y^2 - 2xy$

$4 = x^2 + y^2 - 2 \times 63$

$4 = x^2 + y^2 - 126$

$130 = x^2 + y^2$

Then, $x^2 + y^2 = \boxed { \boxed { 130 } }$

x^2+y^2=(x-y)^2+2xy

=4+63x2

=130