a = 6, is your answer.
Square both sides
9=a−2(6−a)(2a−3)+39=a-2\sqrt{(6-a)(2a-3)}+39=a−2√(6−a)(2a−3)+3
2 .Separate terms with roots from terms without roots
9−a−3=−2(6−a)(2a−3)9-a-3=-2\sqrt{(6-a)(2a-3)}9−a−3=−2√(6−a)(2a−3)
3. Simplify 9−a−39-a-39−a−3 to 6−a6-a6−a
6−a=−2(6−a)(2a−3)6-a=-2\sqrt{(6-a)(2a-3)}6−a=−2√(6−a)(2a−3)
4 .Square both sides
(6−a)2=4(6−a)(2a−3){(6-a)}^{2}=4(6-a)(2a-3)(6−a)2=4(6−a)(2a−3)
5 .Expand
36−12a+a2=48a−72−8a2+12a36-12a+{a}^{2}=48a-72-8{a}^{2}+12a36−12a+a2=48a−72−8a2+12a
6. Simplify 48a−72−8a2+12a48a-72-8{a}^{2}+12a48a−72−8a2+12a to 60a−72−8a260a-72-8{a}^{2}60a−72−8a2
36−12a+a2=60a−72−8a236-12a+{a}^{2}=60a-72-8{a}^{2}36−12a+a2=60a−72−8a2
7. Move all terms to one side
36−12a+a2−60a+72+8a2=036-12a+{a}^{2}-60a+72+8{a}^{2}=036−12a+a2−60a+72+8a2=0
8. Simplify 36−12a+a2−60a+72+8a236-12a+{a}^{2}-60a+72+8{a}^{2}36−12a+a2−60a+72+8a2 to 36−72a+9a2+7236-72a+9{a}^{2}+7236−72a+9a2+72
36−72a+9a2+72=036-72a+9{a}^{2}+72=036−72a+9a2+72=0
9 .Simplify 36−72a+9a2+7236-72a+9{a}^{2}+7236−72a+9a2+72 to −72a+9a2+108-72a+9{a}^{2}+108−72a+9a2+108
−72a+9a2+108=0-72a+9{a}^{2}+108=0−72a+9a2+108=0
10.Factor out the common term 999
−9(8a−a2−12)=0-9(8a-{a}^{2}-12)=0−9(8a−a2−12)=0
11. Factor out the negative sign
−9×−(a2−8a+12)=0-9\times -({a}^{2}-8a+12)=0−9×−(a2−8a+12)=0
12. Divide both sides by −9-9−9
−a2+8a−12=0-{a}^{2}+8a-12=0−a2+8a−12=0
13. Multiply both sides by −1-1−1
a2−8a+12=0{a}^{2}-8a+12=0a2−8a+12=0
14. Factor a2−8a+12{a}^{2}-8a+12a2−8a+12
(a−6)(a−2)=0(a-6)(a-2)=0(a−6)(a−2)=0
15. Solve for aaa
a=6,2a=6,2a=6,2
16 Check solution
When a=2a=2a=2, the original equation −3=6−a−2a−3-3=\sqrt{6-a}-\sqrt{2a-3}−3=√6−a−√2a−3 does not hold true.
We will drop a=2a=2a=2 from the solution set.
17. Therefore,
a=6
