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So I do not know about you but I hate fractions so you can get rid of the fraction -4/3 and 4/9 have a commen denomeator of 9 so if you multiply the equation by 9 or in other words 9(-4/3 x+4/9=-x^2) Then you have 12x+4=-9x^2.  This is a quadratic so you need to set 1 side to 0.  This can be done by adding 9x^2 to both sides.  Then you have 9x^2-12x+4=-9x^2+9x^2.  Then we have 9x^2-12x+4=0.  I learned this cool way to factor a quadratic ax^2+bx+c=0.  You ask yourself what factors of ac add up to give you b.  In this case ac=9(4)=36=-6(-6) and -6-6=-12

Then 9x^2-12x+4=0 becomes

9x^2-6x-6x+4=0 and ask what do the 1st and 2nd have in commen well they have a 3x and ask what do the 3rd and 4th have in commen well they have a  -2 then it becomes 

3x(3x-2)-2(3x-2)=0 Then

(3x-2)(3x-2)=0  Then

(3x-2)^2=0 Then

3x-2=0 Then

3x=2 Then



STEPS 1. Put everything on one side of the equation. a. Add x^2 to each side of the equation (x^2) + -4/3x + 4/9 = -x^2 + (x^2) b. The right side now equals 0. x^2 + -4/3x + 4/9 = 0 2. Try to find a way to solve a. Factoring since it is a quadratic equation b. 4/9= (2^2/3^2)= (2/3)^2 c. - 2/3 + - 2/3= - 4/3 3. Factor. (x - 2/3)(x - 2/3)= 0 4. Use the Zero Product Property and set easy factor equal to zero. (x - 2/3)= 0 (x - 2/3)=0 a. Notice each factor is the same so just sole for one 5. Solve for x (x - 2/3)= 0 a. Add. 2/3 to both sides of the equation. x - 2/3 + 2/3 = 0 + 2/3 b. simplify. x = 2/3 6. Check a. Plug the value you found for x into the original problem to see if you have a solution If the equation is true then you found the solution If the equation is false then that value(x) is not a solution and you can try and other values you found x to be the same way