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x\left(\frac{5}{3}x+2\right)=0
Factor out x.
x=0 x=-\frac{6}{5}
To find equation solutions, solve x=0 and \frac{5x}{3}+2=0.
\frac{5}{3}x^{2}+2x=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.
x=\frac{-2±\sqrt{2^{2}}}{2\times \left(\frac{5}{3}\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{5}{3} for a, 2 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±2}{2\times \left(\frac{5}{3}\right)}
Take the square root of 2^{2}.
x=\frac{-2±2}{\frac{10}{3}}
Multiply 2 times \frac{5}{3}.
x=\frac{0}{\frac{10}{3}}
Now solve the equation x=\frac{-2±2}{\frac{10}{3}} when ± is plus. Add -2 to 2.
x=0
Divide 0 by \frac{10}{3} by multiplying 0 by the reciprocal of \frac{10}{3}.
x=\frac{-4}{\frac{10}{3}}
Now solve the equation x=\frac{-2±2}{\frac{10}{3}} when ± is minus. Subtract 2 from -2.
x=-\frac{6}{5}
Divide -4 by \frac{10}{3} by multiplying -4 by the reciprocal of \frac{10}{3}.
x=0 x=-\frac{6}{5}
The equation is now solved.
\frac{5}{3}x^{2}+2x=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.
\frac{\frac{5}{3}x^{2}+2x}{\frac{5}{3}}=\frac{0}{\frac{5}{3}}
Divide both sides of the equation by \frac{5}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.
x^{2}+\frac{2}{\frac{5}{3}}x=\frac{0}{\frac{5}{3}}
Dividing by \frac{5}{3} undoes the multiplication by \frac{5}{3}.
x^{2}+\frac{6}{5}x=\frac{0}{\frac{5}{3}}
Divide 2 by \frac{5}{3} by multiplying 2 by the reciprocal of \frac{5}{3}.
x^{2}+\frac{6}{5}x=0
Divide 0 by \frac{5}{3} by multiplying 0 by the reciprocal of \frac{5}{3}.
x^{2}+\frac{6}{5}x+\left(\frac{3}{5}\right)^{2}=\left(\frac{3}{5}\right)^{2}
Divide \frac{6}{5}, the coefficient of the x term, by 2 to get \frac{3}{5}. Then add the square of \frac{3}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{6}{5}x+\frac{9}{25}=\frac{9}{25}
Square \frac{3}{5} by squaring both the numerator and the denominator of the fraction.
\left(x+\frac{3}{5}\right)^{2}=\frac{9}{25}
Factor x^{2}+\frac{6}{5}x+\frac{9}{25}. 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{3}{5}\right)^{2}}=\sqrt{\frac{9}{25}}
Take the square root of both sides of the equation.
x+\frac{3}{5}=\frac{3}{5} x+\frac{3}{5}=-\frac{3}{5}
Simplify.
x=0 x=-\frac{6}{5}
Subtract \frac{3}{5} from both sides of the equation.