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r^{2}+\frac{7}{3}r-2=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
r=\frac{-\frac{7}{3}±\sqrt{\left(\frac{7}{3}\right)^{2}-4\left(-2\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, \frac{7}{3} for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
r=\frac{-\frac{7}{3}±\sqrt{\frac{49}{9}-4\left(-2\right)}}{2}
Square \frac{7}{3} by squaring both the numerator and the denominator of the fraction.
r=\frac{-\frac{7}{3}±\sqrt{\frac{49}{9}+8}}{2}
Multiply -4 times -2.
r=\frac{-\frac{7}{3}±\sqrt{\frac{121}{9}}}{2}
Add \frac{49}{9} to 8.
r=\frac{-\frac{7}{3}±\frac{11}{3}}{2}
Take the square root of \frac{121}{9}.
r=\frac{\frac{4}{3}}{2}
Now solve the equation r=\frac{-\frac{7}{3}±\frac{11}{3}}{2} when ± is plus. Add -\frac{7}{3} to \frac{11}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
r=\frac{2}{3}
Divide \frac{4}{3} by 2.
r=-\frac{6}{2}
Now solve the equation r=\frac{-\frac{7}{3}±\frac{11}{3}}{2} when ± is minus. Subtract \frac{11}{3} from -\frac{7}{3} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
r=-3
Divide -6 by 2.
r=\frac{2}{3} r=-3
The equation is now solved.
r^{2}+\frac{7}{3}r-2=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
r^{2}+\frac{7}{3}r-2-\left(-2\right)=-\left(-2\right)
Add 2 to both sides of the equation.
r^{2}+\frac{7}{3}r=-\left(-2\right)
Subtracting -2 from itself leaves 0.
r^{2}+\frac{7}{3}r=2
Subtract -2 from 0.
r^{2}+\frac{7}{3}r+\left(\frac{7}{6}\right)^{2}=2+\left(\frac{7}{6}\right)^{2}
Divide \frac{7}{3}, the coefficient of the x term, by 2 to get \frac{7}{6}. Then add the square of \frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
r^{2}+\frac{7}{3}r+\frac{49}{36}=2+\frac{49}{36}
Square \frac{7}{6} by squaring both the numerator and the denominator of the fraction.
r^{2}+\frac{7}{3}r+\frac{49}{36}=\frac{121}{36}
Add 2 to \frac{49}{36}.
\left(r+\frac{7}{6}\right)^{2}=\frac{121}{36}
Factor r^{2}+\frac{7}{3}r+\frac{49}{36}. 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(r+\frac{7}{6}\right)^{2}}=\sqrt{\frac{121}{36}}
Take the square root of both sides of the equation.
r+\frac{7}{6}=\frac{11}{6} r+\frac{7}{6}=-\frac{11}{6}
Simplify.
r=\frac{2}{3} r=-3
Subtract \frac{7}{6} from both sides of the equation.