Solve 6/x=2x+4/5

$$\frac { 6 } { x } = \frac { 2 x + 4 } { 5 }$$

Solve for x

$$x=-5$$

$$x=3$$

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 5x, the least common multiple of x,5.

$$5\times 6=x\left(2x+4\right)$$

Multiply 5 and 6 to get 30.

$$30=x\left(2x+4\right)$$

Use the distributive property to multiply x by 2x+4.

$$30=2x^{2}+4x$$

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}+4x=30$$

Subtract 30 from both sides.

$$2x^{2}+4x-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 4 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 2\left(-30\right)}}{2\times 2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 2\left(-30\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8\left(-30\right)}}{2\times 2}$$

Multiply -8 times -30.

$$x=\frac{-4±\sqrt{16+240}}{2\times 2}$$

Add 16 to 240.

$$x=\frac{-4±\sqrt{256}}{2\times 2}$$

Take the square root of 256.

$$x=\frac{-4±16}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±16}{4}$$

Now solve the equation x=\frac{-4±16}{4} when ± is plus. Add -4 to 16.

$$x=\frac{12}{4}$$

Divide 12 by 4.

$$x=3$$

Now solve the equation x=\frac{-4±16}{4} when ± is minus. Subtract 16 from -4.

$$x=-\frac{20}{4}$$

Divide -20 by 4.

$$x=-5$$

The equation is now solved.

$$x=3 x=-5$$

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 5x, the least common multiple of x,5.

$$5\times 6=x\left(2x+4\right)$$

Multiply 5 and 6 to get 30.

$$30=x\left(2x+4\right)$$

Use the distributive property to multiply x by 2x+4.

$$30=2x^{2}+4x$$

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}+4x=30$$

Divide both sides by 2.

$$\frac{2x^{2}+4x}{2}=\frac{30}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{4}{2}x=\frac{30}{2}$$

Divide 4 by 2.

$$x^{2}+2x=\frac{30}{2}$$

Divide 30 by 2.

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

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=15+1^{2}$$

Square 1.

$$x^{2}+2x+1=15+1$$

Add 15 to 1.

$$x^{2}+2x+1=16$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{16}$$

Simplify.

$$x+1=4 x+1=-4$$

Subtract 1 from both sides of the equation.

$$x=3 x=-5$$