# A photograph

A photograph will stick to a white square letter with a x cm length. The photo is 3/4 x cm long and 20 cm wide than the width of the paper. The surface of the remaining paper surrounding the photograph is 990 cm2. Find the size of paper and photo.

Correct result:

a =  39.846 cm
b =  29.885 cm
c =  20 cm

#### Solution:

$S_{1}=a^2 \ \\ b=3/4 \cdot \ a \ \\ c=20 \ \\ S_{2}=b \cdot \ c \ \\ S_{1}-S_{2}=990 \ \\ \ \\ a^2 - 3/4*a*20=990 \ \\ \ \\ a^2 - 3/4 \cdot \ a \cdot \ 20=990 \ \\ a^2 -15a -990=0 \ \\ \ \\ p=1; q=-15; r=-990 \ \\ D=q^2 - 4pr=15^2 - 4\cdot 1 \cdot (-990)=4185 \ \\ D>0 \ \\ \ \\ a_{1,2}=\dfrac{ -q \pm \sqrt{ D } }{ 2p }=\dfrac{ 15 \pm \sqrt{ 4185 } }{ 2 }=\dfrac{ 15 \pm 3 \sqrt{ 465 } }{ 2 } \ \\ a_{1,2}=7.5 \pm 32.3457879793 \ \\ a_{1}=39.8457879793 \ \\ a_{2}=-24.8457879793 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (a -39.8457879793) (a +24.8457879793)=0 \ \\ \ \\ a=a_{1}=39.8458=39.846 \ \text{cm}$

Checkout calculation with our calculator of quadratic equations.

$b=3/4 \cdot \ a=3/4 \cdot \ 39.8458=29.885 \ \text{cm}$
$c=20 \ \text{cm} \ \\ \ \\ S_{1}=a^2=39.8458^2 \doteq 1587.7037 \ \text{cm}^2 \ \\ S_{2}=b \cdot \ c=29.8845 \cdot \ 20=\dfrac{ 5977 }{ 10 }=597.7 \ \text{cm}^2$

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