Solve x+5=20/x-3

$$x + 5 = \frac { 20 } { x } - 3$$

Solve for x

$$x=-10$$

$$x=2$$

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.

$$xx+x\times 5=20+x\left(-3\right)$$

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

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

Subtract 20 from both sides.

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

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

$$x^{2}+x\times 5-20-x\left(-3\right)=0$$

Combine x\times 5 and -x\left(-3\right) to get 8x.

$$x^{2}+8x-20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-20\right)}}{2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-20\right)}}{2}$$

Multiply -4 times -20.

$$x=\frac{-8±\sqrt{64+80}}{2}$$

Add 64 to 80.

$$x=\frac{-8±\sqrt{144}}{2}$$

Take the square root of 144.

$$x=\frac{-8±12}{2}$$

Now solve the equation x=\frac{-8±12}{2} when ± is plus. Add -8 to 12.

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

Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{-8±12}{2} when ± is minus. Subtract 12 from -8.

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

Divide -20 by 2.

$$x=-10$$

The equation is now solved.

$$x=2 x=-10$$

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.

$$xx+x\times 5=20+x\left(-3\right)$$

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

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

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

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

Combine x\times 5 and -x\left(-3\right) to get 8x.

$$x^{2}+8x=20$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=20+4^{2}$$

Square 4.

$$x^{2}+8x+16=20+16$$

Add 20 to 16.

$$x^{2}+8x+16=36$$

Factor x^{2}+8x+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}.

$$\left(x+4\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{36}$$

Simplify.

$$x+4=6 x+4=-6$$

Subtract 4 from both sides of the equation.

$$x=2 x=-10$$