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Find the value of x in the following figure Pls give a detailed solution

Find the value of x in the following figure 
Pls give a detailed solution
 

Question Image
Grade:6

2 Answers

Leena
10 Points
3 years ago
 Given
  AB,AC,AD,BC,BE,CD are straight lines.
 
AE=x, BE=CD=x−3, BC=10, AD=x+4
∠BEC=90∘
Find x .............................
Kushal Chaudhari
102 Points
3 years ago
Dear leena hope this helps
By the Pythagorean theorem we have
 
CE=102−(x−3)2−−−−−−−−−−−√=91−x2+6x−−−−−−−−−−√
and
CD2+AD2=AC2=(CE+AE)2.
So we have to solve the following irrational equation
(x−3)2+(x+4)2=(91−x2+6x−−−−−−−−−−√+x)2,(1)
which can be simplified to
x2−2x−33=−x4+6x3+91x2−−−−−−−−−−−−−−√.
After squaring both sides and grouping the terms of the same degree we get the quartic equation
2x4−10x3−153x2+132x+1089=0.(2)
The coefficient of x4 is 2=1×2 and the constant term is 1089=1×32112. To find possible rational roots of this equation, we apply the rational root theorem and test the numbers of the form
x=±pq,
where p∈{1,3,9,11,33,99,121,363,1089} is a divisor of 1089 and q∈{1,2} is a divisor of 2. It turns out that x=3 and x=11 are roots. Now we divide the LHS by x−3
2x4−10x3−153x2+132x+1089x−3=2x3−4x2−165x−363
and this quotient by x−11
2x3−4x2−165x−363x−11=2x2+18x+33.
So we have the equivalent equation
(x−3)(x−11)(2x2+18x+33)=0(3)
Since the solutions of 2x2+18x+33 are both negative and x=3 is not a solution of the original irrational equation, the solution is therefore
x=11.

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