Solve 9/x-3-27/x(x-3)=-3

$$\frac { 9 } { x - 3 } - \frac { 27 } { x ( x - 3 ) } = - 3$$

Solve for x

$$x=-3$$

Steps Using Difference of Squares

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x-3,x\left(x-3\right).

$$x\times 9-27=-3x\left(x-3\right)$$

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

$$x\times 9-27=-3x^{2}+9x$$

Add 3x^{2} to both sides.

$$x\times 9-27+3x^{2}=9x$$

Subtract 9x from both sides.

$$x\times 9-27+3x^{2}-9x=0$$

Combine x\times 9 and -9x to get 0.

$$-27+3x^{2}=0$$

Divide both sides by 3.

$$-9+x^{2}=0$$

Consider -9+x^{2}. Rewrite -9+x^{2} as x^{2}-3^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-3\right)\left(x+3\right)=0$$

To find equation solutions, solve x-3=0 and x+3=0.

$$x=3 x=-3$$

Variable x cannot be equal to 3.

$$x=-3$$

Steps by Finding Square Root

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x-3,x\left(x-3\right).

$$x\times 9-27=-3x\left(x-3\right)$$

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

$$x\times 9-27=-3x^{2}+9x$$

Add 3x^{2} to both sides.

$$x\times 9-27+3x^{2}=9x$$

Subtract 9x from both sides.

$$x\times 9-27+3x^{2}-9x=0$$

Combine x\times 9 and -9x to get 0.

$$-27+3x^{2}=0$$

Add 27 to both sides. Anything plus zero gives itself.

$$3x^{2}=27$$

Divide both sides by 3.

$$x^{2}=\frac{27}{3}$$

Divide 27 by 3 to get 9.

$$x^{2}=9$$

Take the square root of both sides of the equation.

$$x=3 x=-3$$

Variable x cannot be equal to 3.

$$x=-3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x-3,x\left(x-3\right).

$$x\times 9-27=-3x\left(x-3\right)$$

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

$$x\times 9-27=-3x^{2}+9x$$

Add 3x^{2} to both sides.

$$x\times 9-27+3x^{2}=9x$$

Subtract 9x from both sides.

$$x\times 9-27+3x^{2}-9x=0$$

Combine x\times 9 and -9x to get 0.

$$-27+3x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$3x^{2}-27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 0 for b, and -27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 3\left(-27\right)}}{2\times 3}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 3\left(-27\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{0±\sqrt{-12\left(-27\right)}}{2\times 3}$$

Multiply -12 times -27.

$$x=\frac{0±\sqrt{324}}{2\times 3}$$

Take the square root of 324.

$$x=\frac{0±18}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{0±18}{6}$$

Now solve the equation x=\frac{0±18}{6} when ± is plus. Divide 18 by 6.

$$x=3$$

Now solve the equation x=\frac{0±18}{6} when ± is minus. Divide -18 by 6.

$$x=-3$$

The equation is now solved.

$$x=3 x=-3$$

Variable x cannot be equal to 3.

$$x=-3$$