# Diamond diagonals

Calculate the diamonds' diagonals lengths if the diamond area is 156 cm square and the side length is 13 cm.

Correct result:

u =  21.633 cm
v =  14.422 cm

#### Solution:

$S=156 \ \text{cm}^2 \ \\ a=13 \ \text{cm} \ \\ S=uv/2 \ \\ a^2=(u/2)^2+(v/2)^2 \ \\ 4a^2=u^2 + v^2 \ \\ 676=u^2+v^2 \ \\ 312=uv \ \\ v=312/u \ \\ 676=u^2+(312/u)^2 \ \\ x=u^2 \ \\ 676=x+312^2/x \ \\ \ \\ 676x=x^2+312^2 \ \\ \ \\ -x^2 +676x -97344=0 \ \\ x^2 -676x +97344=0 \ \\ \ \\ a=1; b=-676; c=97344 \ \\ D=b^2 - 4ac=676^2 - 4\cdot 1 \cdot 97344=67600 \ \\ D>0 \ \\ \ \\ x_{1,2}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ 676 \pm \sqrt{ 67600 } }{ 2 } \ \\ x_{1,2}=\dfrac{ 676 \pm 260 }{ 2 } \ \\ x_{1,2}=338 \pm 130 \ \\ x_{1}=468 \ \\ x_{2}=208 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (x -468) (x -208)=0 \ \\ \ \\ u=\sqrt{ x_{1} }=\sqrt{ 468 }=6 \ \sqrt{ 13 }=21.633 \ \text{cm}$

Checkout calculation with our calculator of quadratic equations.

$v=\sqrt{ x_{2} }=\sqrt{ 208 }=4 \ \sqrt{ 13 }=14.422 \ \text{cm}$

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