Solve 3(x-2)/4-20x-5/x=46

$$\frac { 3 ( x - 2 ) } { 4 } - \frac { 20 x - 5 } { x } = 46$$

Solve for x

$$x=\frac{\sqrt{18165}}{3}+45\approx 89.925864859$$

$$x=-\frac{\sqrt{18165}}{3}+45\approx 0.074135141$$

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 4x, the least common multiple of 4,x.

$$x\times 3\left(x-2\right)-4\left(20x-5\right)=184x$$

Use the distributive property to multiply x\times 3 by x-2.

$$3x^{2}-2x\times 3-4\left(20x-5\right)=184x$$

Multiply -2 and 3 to get -6.

$$3x^{2}-6x-4\left(20x-5\right)=184x$$

Use the distributive property to multiply -4 by 20x-5.

$$3x^{2}-6x-80x+20=184x$$

Combine -6x and -80x to get -86x.

$$3x^{2}-86x+20=184x$$

Subtract 184x from both sides.

$$3x^{2}-86x+20-184x=0$$

Combine -86x and -184x to get -270x.

$$3x^{2}-270x+20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -270 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-270\right)±\sqrt{\left(-270\right)^{2}-4\times 3\times 20}}{2\times 3}$$

Square -270.

$$x=\frac{-\left(-270\right)±\sqrt{72900-4\times 3\times 20}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-270\right)±\sqrt{72900-12\times 20}}{2\times 3}$$

Multiply -12 times 20.

$$x=\frac{-\left(-270\right)±\sqrt{72900-240}}{2\times 3}$$

Add 72900 to -240.

$$x=\frac{-\left(-270\right)±\sqrt{72660}}{2\times 3}$$

Take the square root of 72660.

$$x=\frac{-\left(-270\right)±2\sqrt{18165}}{2\times 3}$$

The opposite of -270 is 270.

$$x=\frac{270±2\sqrt{18165}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{270±2\sqrt{18165}}{6}$$

Now solve the equation x=\frac{270±2\sqrt{18165}}{6} when ± is plus. Add 270 to 2\sqrt{18165}.

$$x=\frac{2\sqrt{18165}+270}{6}$$

Divide 270+2\sqrt{18165} by 6.

$$x=\frac{\sqrt{18165}}{3}+45$$

Now solve the equation x=\frac{270±2\sqrt{18165}}{6} when ± is minus. Subtract 2\sqrt{18165} from 270.

$$x=\frac{270-2\sqrt{18165}}{6}$$

Divide 270-2\sqrt{18165} by 6.

$$x=-\frac{\sqrt{18165}}{3}+45$$

The equation is now solved.

$$x=\frac{\sqrt{18165}}{3}+45 x=-\frac{\sqrt{18165}}{3}+45$$

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 4x, the least common multiple of 4,x.

$$x\times 3\left(x-2\right)-4\left(20x-5\right)=184x$$

Use the distributive property to multiply x\times 3 by x-2.

$$3x^{2}-2x\times 3-4\left(20x-5\right)=184x$$

Multiply -2 and 3 to get -6.

$$3x^{2}-6x-4\left(20x-5\right)=184x$$

Use the distributive property to multiply -4 by 20x-5.

$$3x^{2}-6x-80x+20=184x$$

Combine -6x and -80x to get -86x.

$$3x^{2}-86x+20=184x$$

Subtract 184x from both sides.

$$3x^{2}-86x+20-184x=0$$

Combine -86x and -184x to get -270x.

$$3x^{2}-270x+20=0$$

Subtract 20 from both sides. Anything subtracted from zero gives its negation.

$$3x^{2}-270x=-20$$

Divide both sides by 3.

$$\frac{3x^{2}-270x}{3}=-\frac{20}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{270}{3}\right)x=-\frac{20}{3}$$

Divide -270 by 3.

$$x^{2}-90x=-\frac{20}{3}$$

Divide -90, the coefficient of the x term, by 2 to get -45. Then add the square of -45 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-90x+\left(-45\right)^{2}=-\frac{20}{3}+\left(-45\right)^{2}$$

Square -45.

$$x^{2}-90x+2025=-\frac{20}{3}+2025$$

Add -\frac{20}{3} to 2025.

$$x^{2}-90x+2025=\frac{6055}{3}$$

Factor x^{2}-90x+2025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-45\right)^{2}=\frac{6055}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-45\right)^{2}}=\sqrt{\frac{6055}{3}}$$

Simplify.

$$x-45=\frac{\sqrt{18165}}{3} x-45=-\frac{\sqrt{18165}}{3}$$

Add 45 to both sides of the equation.

$$x=\frac{\sqrt{18165}}{3}+45 x=-\frac{\sqrt{18165}}{3}+45$$