Answer:
The solution is x =2 for the given equation [tex]\sqrt{9x+7} +\sqrt{2x}=7[/tex]
Step-by-step explanation:
We need to find the solution for x in the given equation.
[tex]\sqrt{9x+7} +\sqrt{2x}=7[/tex]
Solving:
Subtract [tex]\sqrt{2x}[/tex] from both sides
[tex]\sqrt{9x+7} =7-\sqrt{2x}[/tex]
Taking square on both sides
[tex](\sqrt{9x+7})^2 =(7-\sqrt{2x})^2\\Using\,\, (a-b)^2 = a^2-2ab-b^2\\9x+7=(7)^2-2(7)(\sqrt{2x})+(\sqrt{2x})^2\\9x+7=49-14\sqrt{2x}+2x\\9x-2x+7-49=-14\sqrt{2x}\\7x-42=-14\sqrt{2x}\\Taking\,\,square\,\,on\,\,both\,\,sides\\(7x-42)^2=(-14\sqrt{2x})^2\\49x^2-2(7x)(42)+(42)^2= 196(2x)\\49x^2-588x+1764=392x\\49x^2-588x+1764-392x=0\\49x^2-980x+1764=0\\Using \,\,quadratic \,\, equation \,\,to \,\, find \,\, value \,\, of \,\, x \\x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\a= 49, \, b= -980 \,\,and\,\, c = 1764[/tex]
Putting values and solving
[tex]x=\frac{-(-980)\pm\sqrt{(-980)^2-4(49)(1760)}}{2(49)}\\x=\frac{980\pm\sqrt{614656}}{98}\\Solving\\x=18 \,\, and x =2[/tex]
Verifying the solution by putting values of x in given equation
Putting x=18
[tex]\sqrt{9x+7} +\sqrt{2x}=7\\\sqrt{9(18)+7} +\sqrt{2(18)}=7\\\sqrt{169} +\sqrt{36}=7\\13+6=7\\19\neq 7[/tex]
So, x=18 is not solution o given equation.
Putting x = 2
[tex]\sqrt{9x+7} +\sqrt{2x}=7\\\sqrt{9(2)+7} +\sqrt{2(2)}=7\\\sqrt{25} +\sqrt{4}=7\\5+2=7\\7=7[/tex]
So, x=2 satisfies the equation
The solution is x =2 for the given equation [tex]\sqrt{9x+7} +\sqrt{2x}=7[/tex]
