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3x\left(3x+2\right)+\left(3x+2\right)\times 2+1=7\left(3x+2\right)
Variable x cannot be equal to -\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by 3x+2.
9x^{2}+6x+\left(3x+2\right)\times 2+1=7\left(3x+2\right)
Use the distributive property to multiply 3x by 3x+2.
9x^{2}+6x+6x+4+1=7\left(3x+2\right)
Use the distributive property to multiply 3x+2 by 2.
9x^{2}+12x+4+1=7\left(3x+2\right)
Combine 6x and 6x to get 12x.
9x^{2}+12x+5=7\left(3x+2\right)
Add 4 and 1 to get 5.
9x^{2}+12x+5=21x+14
Use the distributive property to multiply 7 by 3x+2.
9x^{2}+12x+5-21x=14
Subtract 21x from both sides.
9x^{2}-9x+5=14
Combine 12x and -21x to get -9x.
9x^{2}-9x+5-14=0
Subtract 14 from both sides.
9x^{2}-9x-9=0
Subtract 14 from 5 to get -9.
x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 9\left(-9\right)}}{2\times 9}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -9 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-9\right)±\sqrt{81-4\times 9\left(-9\right)}}{2\times 9}
Square -9.
x=\frac{-\left(-9\right)±\sqrt{81-36\left(-9\right)}}{2\times 9}
Multiply -4 times 9.
x=\frac{-\left(-9\right)±\sqrt{81+324}}{2\times 9}
Multiply -36 times -9.
x=\frac{-\left(-9\right)±\sqrt{405}}{2\times 9}
Add 81 to 324.
x=\frac{-\left(-9\right)±9\sqrt{5}}{2\times 9}
Take the square root of 405.
x=\frac{9±9\sqrt{5}}{2\times 9}
The opposite of -9 is 9.
x=\frac{9±9\sqrt{5}}{18}
Multiply 2 times 9.
x=\frac{9\sqrt{5}+9}{18}
Now solve the equation x=\frac{9±9\sqrt{5}}{18} when ± is plus. Add 9 to 9\sqrt{5}.
x=\frac{\sqrt{5}+1}{2}
Divide 9+9\sqrt{5} by 18.
x=\frac{9-9\sqrt{5}}{18}
Now solve the equation x=\frac{9±9\sqrt{5}}{18} when ± is minus. Subtract 9\sqrt{5} from 9.
x=\frac{1-\sqrt{5}}{2}
Divide 9-9\sqrt{5} by 18.
x=\frac{\sqrt{5}+1}{2} x=\frac{1-\sqrt{5}}{2}
The equation is now solved.
3x\left(3x+2\right)+\left(3x+2\right)\times 2+1=7\left(3x+2\right)
Variable x cannot be equal to -\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by 3x+2.
9x^{2}+6x+\left(3x+2\right)\times 2+1=7\left(3x+2\right)
Use the distributive property to multiply 3x by 3x+2.
9x^{2}+6x+6x+4+1=7\left(3x+2\right)
Use the distributive property to multiply 3x+2 by 2.
9x^{2}+12x+4+1=7\left(3x+2\right)
Combine 6x and 6x to get 12x.
9x^{2}+12x+5=7\left(3x+2\right)
Add 4 and 1 to get 5.
9x^{2}+12x+5=21x+14
Use the distributive property to multiply 7 by 3x+2.
9x^{2}+12x+5-21x=14
Subtract 21x from both sides.
9x^{2}-9x+5=14
Combine 12x and -21x to get -9x.
9x^{2}-9x=14-5
Subtract 5 from both sides.
9x^{2}-9x=9
Subtract 5 from 14 to get 9.
\frac{9x^{2}-9x}{9}=\frac{9}{9}
Divide both sides by 9.
x^{2}+\left(-\frac{9}{9}\right)x=\frac{9}{9}
Dividing by 9 undoes the multiplication by 9.
x^{2}-x=\frac{9}{9}
Divide -9 by 9.
x^{2}-x=1
Divide 9 by 9.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=1+\left(-\frac{1}{2}\right)^{2}
Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-x+\frac{1}{4}=1+\frac{1}{4}
Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-x+\frac{1}{4}=\frac{5}{4}
Add 1 to \frac{1}{4}.
\left(x-\frac{1}{2}\right)^{2}=\frac{5}{4}
Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{5}{4}}
Take the square root of both sides of the equation.
x-\frac{1}{2}=\frac{\sqrt{5}}{2} x-\frac{1}{2}=-\frac{\sqrt{5}}{2}
Simplify.
x=\frac{\sqrt{5}+1}{2} x=\frac{1-\sqrt{5}}{2}
Add \frac{1}{2} to both sides of the equation.