Solve for x

x = -\frac{6}{5} = -1\frac{1}{5} = -1.2

x=0

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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.

Examples

Quadratic equation

{ x } ^ { 2 } - 4 x - 5 = 0

Trigonometry

4 \sin \theta \cos \theta = 2 \sin \theta

Linear equation

y = 3x + 4

Arithmetic

699 * 533

Matrix

\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]

Simultaneous equation

\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.

Differentiation

\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }

Integration

\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x

Limits

\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}