If the value of $xy$ , If $ x^2 + y^2 = 34$ and $x + y =8$

Question

If $ x^2 + y^2 = 34$ and $x + y =8$ find the value of xy

So what should I be considering here the sum of two squares or..

I'm struggling here & any help to figure this out will be kindly appreciated

Answer 1

HINT: Square the second equation.

Answer 2

Use:

$$x+y=8\Longleftrightarrow x=8-y\Longleftrightarrow x^2=\left(8-y\right)^2$$

So:

$$x^2+y^2=34\Longleftrightarrow\left(8-y\right)^2+y^2=34$$

So, for $y$ we find $y=3$ or $y=5$

Answer 3

$x^2$ + $y^2$ = $34$ $-->$ Equation 1

$x + y = 8$ $-->$ Equation 2

Now, we can rewrite Equation 2 with respect to x. This value will be substituted into Equation 1.

$x = 8 - y$

$x^2 + y^2 = 34$

$(8 - y)^2 + y^2 = 34$

$64 - 16y + y^2 + y^2 = 34$

$64 - 16y + 2y^2 = 34$

$2y^2 - 16y + 30 = 0$

Solving the quadratic above, the roots are: $$y1 = 5$$ and $$y2 = 3$$

Now plug both of these values into Equation 2. This will help you find the corresponding x-value. Once this is obtained, multiply the two numbers.

$x + y = 8$

$x + 5 = 8$

$x = 3$

$$xy = 3(5) = 15$$

This is the first solution.

$x + y = 8$

$x + 3 = 8$

$x = 5$

$$xy = 5(3) = 15$$

In both cases, the solution is 15, and this is due to the symmetry of the two points. Therefore, xy must be equal to 15.