# The sum of the squares of two negative numbers is 106 and the difference of the squares of the number is 56 find the numbers

The sum of the squares of two negative numbers is 106 and the difference of the squares of the number is 56 find the numbers
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Clara Reese

Calculation:
Let two negative numbers are x and y.
The sum of the square of two negative numbers is 106.
${x}^{2}+{y}^{2}=106$ (1)
And the difference of the square of the number is 56.
${x}^{2}-{y}^{2}=56$ (2)
$\left({x}^{2}+{y}^{2}\right)+\left({x}^{2}-{y}^{2}\right)=106+56$
$2{x}^{2}+0=162$
$2{x}^{2}=162$
$\left(2{x}^{2}\right)/2=162/2$ [Divide by 2]
${x}^{2}=81$
Taking square root.
$\sqrt{{x}^{2}}=±\sqrt{81}$
$x=±\sqrt{{9}^{2}}$
$x=±9$
x=-9, because x is negative
Substitute x = -9 in equation (1).
$-{9}^{2}+{y}^{2}=106$
$81+{y}^{2}=106$
$-81+81+{y}^{2}=106-81$ [Substract 81 from both sides]
$={y}^{2}=25$
Taking square root.
$\sqrt{{y}^{2}}=±\sqrt{25}$
$y=\sqrt{{5}^{2}}$
$y=±5$
y=-5, because y is negative
Thus, the negative numbers are x = -9 and y = -5.