Solve 2(x-3)=20/x-3

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

Solve for x

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$x=4$$

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 x.

$$2\left(x-3\right)x=20+x\left(-3\right)$$

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

$$\left(2x-6\right)x=20+x\left(-3\right)$$

Use the distributive property to multiply 2x-6 by x.

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

Subtract 20 from both sides.

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

Subtract x\left(-3\right) from both sides.

$$2x^{2}-6x-20-x\left(-3\right)=0$$

Combine -6x and -x\left(-3\right) to get -3x.

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

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

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

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\left(-20\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-3\right)±\sqrt{9-8\left(-20\right)}}{2\times 2}$$

Multiply -8 times -20.

$$x=\frac{-\left(-3\right)±\sqrt{9+160}}{2\times 2}$$

Add 9 to 160.

$$x=\frac{-\left(-3\right)±\sqrt{169}}{2\times 2}$$

Take the square root of 169.

$$x=\frac{-\left(-3\right)±13}{2\times 2}$$

The opposite of -3 is 3.

$$x=\frac{3±13}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{3±13}{4}$$

Now solve the equation x=\frac{3±13}{4} when ± is plus. Add 3 to 13.

$$x=\frac{16}{4}$$

Divide 16 by 4.

$$x=4$$

Now solve the equation x=\frac{3±13}{4} when ± is minus. Subtract 13 from 3.

$$x=-\frac{10}{4}$$

Reduce the fraction \frac{-10}{4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{5}{2}$$

The equation is now solved.

$$x=4 x=-\frac{5}{2}$$

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 x.

$$2\left(x-3\right)x=20+x\left(-3\right)$$

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

$$\left(2x-6\right)x=20+x\left(-3\right)$$

Use the distributive property to multiply 2x-6 by x.

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

Subtract x\left(-3\right) from both sides.

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

Combine -6x and -x\left(-3\right) to get -3x.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide 20 by 2.

$$x^{2}-\frac{3}{2}x=10$$

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=10+\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=10+\frac{9}{16}$$

Add 10 to \frac{9}{16}.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{169}{16}$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{169}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{169}{16}}$$

Simplify.

$$x-\frac{3}{4}=\frac{13}{4} x-\frac{3}{4}=-\frac{13}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$x=4 x=-\frac{5}{2}$$