Solve 5x/2x=2x/3x^2

$$\frac { 5 x } { 2 x } = \frac { 2 x } { 3 x ^ { 2 } }$$

Solve for x

$$x=\frac{4}{15}\approx 0.266666667$$

Steps Using Factoring

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6x^{2}, the least common multiple of 2x,3x^{2}.

$$3x\times 5x=2\times 2x$$

Multiply 3 and 5 to get 15.

$$15xx=2\times 2x$$

Multiply x and x to get x^{2}.

$$15x^{2}=2\times 2x$$

Multiply 2 and 2 to get 4.

$$15x^{2}=4x$$

Subtract 4x from both sides.

$$15x^{2}-4x=0$$

Factor out x.

$$x\left(15x-4\right)=0$$

To find equation solutions, solve x=0 and 15x-4=0.

$$x=0 x=\frac{4}{15}$$

Variable x cannot be equal to 0.

$$x=\frac{4}{15}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6x^{2}, the least common multiple of 2x,3x^{2}.

$$3x\times 5x=2\times 2x$$

Multiply 3 and 5 to get 15.

$$15xx=2\times 2x$$

Multiply x and x to get x^{2}.

$$15x^{2}=2\times 2x$$

Multiply 2 and 2 to get 4.

$$15x^{2}=4x$$

Subtract 4x from both sides.

$$15x^{2}-4x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, -4 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}}}{2\times 15}$$

Take the square root of \left(-4\right)^{2}.

$$x=\frac{-\left(-4\right)±4}{2\times 15}$$

The opposite of -4 is 4.

$$x=\frac{4±4}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{4±4}{30}$$

Now solve the equation x=\frac{4±4}{30} when ± is plus. Add 4 to 4.

$$x=\frac{8}{30}$$

Reduce the fraction \frac{8}{30} to lowest terms by extracting and canceling out 2.

$$x=\frac{4}{15}$$

Now solve the equation x=\frac{4±4}{30} when ± is minus. Subtract 4 from 4.

$$x=\frac{0}{30}$$

Divide 0 by 30.

$$x=0$$

The equation is now solved.

$$x=\frac{4}{15} x=0$$

Variable x cannot be equal to 0.

$$x=\frac{4}{15}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6x^{2}, the least common multiple of 2x,3x^{2}.

$$3x\times 5x=2\times 2x$$

Multiply 3 and 5 to get 15.

$$15xx=2\times 2x$$

Multiply x and x to get x^{2}.

$$15x^{2}=2\times 2x$$

Multiply 2 and 2 to get 4.

$$15x^{2}=4x$$

Subtract 4x from both sides.

$$15x^{2}-4x=0$$

Divide both sides by 15.

$$\frac{15x^{2}-4x}{15}=\frac{0}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$x^{2}-\frac{4}{15}x=\frac{0}{15}$$

Divide 0 by 15.

$$x^{2}-\frac{4}{15}x=0$$

Divide -\frac{4}{15}, the coefficient of the x term, by 2 to get -\frac{2}{15}. Then add the square of -\frac{2}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4}{15}x+\left(-\frac{2}{15}\right)^{2}=\left(-\frac{2}{15}\right)^{2}$$

Square -\frac{2}{15} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4}{15}x+\frac{4}{225}=\frac{4}{225}$$

Factor x^{2}-\frac{4}{15}x+\frac{4}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2}{15}\right)^{2}=\frac{4}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2}{15}\right)^{2}}=\sqrt{\frac{4}{225}}$$

Simplify.

$$x-\frac{2}{15}=\frac{2}{15} x-\frac{2}{15}=-\frac{2}{15}$$

Add \frac{2}{15} to both sides of the equation.

$$x=\frac{4}{15} x=0$$

Variable x cannot be equal to 0.

$$x=\frac{4}{15}$$