Solve 5x/x-3=5/x^2-x-6

$$\frac { 5 x } { x - 3 } = \frac { 5 } { x ^ { 2 } - x - 6 }$$

Solve for x (complex solution)

$$x=\sqrt{2}-1\approx 0.414213562$$

$$x=-\left(\sqrt{2}+1\right)\approx -2.414213562$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+2\right), the least common multiple of x-3,x^{2}-x-6.

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

Use the distributive property to multiply x+2 by 5.

$$\left(5x+10\right)x=5$$

Use the distributive property to multiply 5x+10 by x.

$$5x^{2}+10x=5$$

Subtract 5 from both sides.

$$5x^{2}+10x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 10 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 5\left(-5\right)}}{2\times 5}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\times 5\left(-5\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-10±\sqrt{100-20\left(-5\right)}}{2\times 5}$$

Multiply -20 times -5.

$$x=\frac{-10±\sqrt{100+100}}{2\times 5}$$

Add 100 to 100.

$$x=\frac{-10±\sqrt{200}}{2\times 5}$$

Take the square root of 200.

$$x=\frac{-10±10\sqrt{2}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-10±10\sqrt{2}}{10}$$

Now solve the equation x=\frac{-10±10\sqrt{2}}{10} when ± is plus. Add -10 to 10\sqrt{2}.

$$x=\frac{10\sqrt{2}-10}{10}$$

Divide -10+10\sqrt{2} by 10.

$$x=\sqrt{2}-1$$

Now solve the equation x=\frac{-10±10\sqrt{2}}{10} when ± is minus. Subtract 10\sqrt{2} from -10.

$$x=\frac{-10\sqrt{2}-10}{10}$$

Divide -10-10\sqrt{2} by 10.

$$x=-\sqrt{2}-1$$

The equation is now solved.

$$x=\sqrt{2}-1 x=-\sqrt{2}-1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+2\right), the least common multiple of x-3,x^{2}-x-6.

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

Use the distributive property to multiply x+2 by 5.

$$\left(5x+10\right)x=5$$

Use the distributive property to multiply 5x+10 by x.

$$5x^{2}+10x=5$$

Divide both sides by 5.

$$\frac{5x^{2}+10x}{5}=\frac{5}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{10}{5}x=\frac{5}{5}$$

Divide 10 by 5.

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

Divide 5 by 5.

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

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.

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

Square 1.

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

Add 1 to 1.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+1=\sqrt{2} x+1=-\sqrt{2}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{2}-1 x=-\sqrt{2}-1$$

Solve for x

$$x=\sqrt{2}-1\approx 0.414213562$$

$$x=-\sqrt{2}-1\approx -2.414213562$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+2\right), the least common multiple of x-3,x^{2}-x-6.

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

Use the distributive property to multiply x+2 by 5.

$$\left(5x+10\right)x=5$$

Use the distributive property to multiply 5x+10 by x.

$$5x^{2}+10x=5$$

Subtract 5 from both sides.

$$5x^{2}+10x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 10 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 5\left(-5\right)}}{2\times 5}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\times 5\left(-5\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-10±\sqrt{100-20\left(-5\right)}}{2\times 5}$$

Multiply -20 times -5.

$$x=\frac{-10±\sqrt{100+100}}{2\times 5}$$

Add 100 to 100.

$$x=\frac{-10±\sqrt{200}}{2\times 5}$$

Take the square root of 200.

$$x=\frac{-10±10\sqrt{2}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-10±10\sqrt{2}}{10}$$

Now solve the equation x=\frac{-10±10\sqrt{2}}{10} when ± is plus. Add -10 to 10\sqrt{2}.

$$x=\frac{10\sqrt{2}-10}{10}$$

Divide -10+10\sqrt{2} by 10.

$$x=\sqrt{2}-1$$

Now solve the equation x=\frac{-10±10\sqrt{2}}{10} when ± is minus. Subtract 10\sqrt{2} from -10.

$$x=\frac{-10\sqrt{2}-10}{10}$$

Divide -10-10\sqrt{2} by 10.

$$x=-\sqrt{2}-1$$

The equation is now solved.

$$x=\sqrt{2}-1 x=-\sqrt{2}-1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+2\right), the least common multiple of x-3,x^{2}-x-6.

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

Use the distributive property to multiply x+2 by 5.

$$\left(5x+10\right)x=5$$

Use the distributive property to multiply 5x+10 by x.

$$5x^{2}+10x=5$$

Divide both sides by 5.

$$\frac{5x^{2}+10x}{5}=\frac{5}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{10}{5}x=\frac{5}{5}$$

Divide 10 by 5.

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

Divide 5 by 5.

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

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.

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

Square 1.

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

Add 1 to 1.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+1=\sqrt{2} x+1=-\sqrt{2}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{2}-1 x=-\sqrt{2}-1$$