Solve 4/x-3+2/x=1

$$\frac { 4 } { x - 3 } + \frac { 2 } { x } = 1$$

Solve for x

$$x = \frac{\sqrt{57} + 9}{2} \approx 8.274917218$$

$$x=\frac{9-\sqrt{57}}{2}\approx 0.725082782$$

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.

$$x\times 4+\left(x-3\right)\times 2=x\left(x-3\right)$$

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

$$x\times 4+2x-6=x\left(x-3\right)$$

Combine x\times 4 and 2x to get 6x.

$$6x-6=x\left(x-3\right)$$

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

$$6x-6=x^{2}-3x$$

Subtract x^{2} from both sides.

$$6x-6-x^{2}=-3x$$

Add 3x to both sides.

$$6x-6-x^{2}+3x=0$$

Combine 6x and 3x to get 9x.

$$9x-6-x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+9x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 9 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-9±\sqrt{81+4\left(-6\right)}}{2\left(-1\right)}$$

Multiply 4 times -6.

$$x=\frac{-9±\sqrt{81-24}}{2\left(-1\right)}$$

Add 81 to -24.

$$x=\frac{-9±\sqrt{57}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-9±\sqrt{57}}{-2}$$

Now solve the equation x=\frac{-9±\sqrt{57}}{-2} when ± is plus. Add -9 to \sqrt{57}.

$$x=\frac{\sqrt{57}-9}{-2}$$

Divide -9+\sqrt{57} by -2.

$$x=\frac{9-\sqrt{57}}{2}$$

Now solve the equation x=\frac{-9±\sqrt{57}}{-2} when ± is minus. Subtract \sqrt{57} from -9.

$$x=\frac{-\sqrt{57}-9}{-2}$$

Divide -9-\sqrt{57} by -2.

$$x=\frac{\sqrt{57}+9}{2}$$

The equation is now solved.

$$x=\frac{9-\sqrt{57}}{2} x=\frac{\sqrt{57}+9}{2}$$

Steps for Completing the Square

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.

$$x\times 4+\left(x-3\right)\times 2=x\left(x-3\right)$$

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

$$x\times 4+2x-6=x\left(x-3\right)$$

Combine x\times 4 and 2x to get 6x.

$$6x-6=x\left(x-3\right)$$

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

$$6x-6=x^{2}-3x$$

Subtract x^{2} from both sides.

$$6x-6-x^{2}=-3x$$

Add 3x to both sides.

$$6x-6-x^{2}+3x=0$$

Combine 6x and 3x to get 9x.

$$9x-6-x^{2}=0$$

Add 6 to both sides. Anything plus zero gives itself.

$$9x-x^{2}=6$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}+9x=6$$

Divide both sides by -1.

$$\frac{-x^{2}+9x}{-1}=\frac{6}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{9}{-1}x=\frac{6}{-1}$$

Divide 9 by -1.

$$x^{2}-9x=\frac{6}{-1}$$

Divide 6 by -1.

$$x^{2}-9x=-6$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-6+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=-6+\frac{81}{4}$$

Add -6 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{57}{4}$$

Factor x^{2}-9x+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{57}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{57}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{\sqrt{57}}{2} x-\frac{9}{2}=-\frac{\sqrt{57}}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{\sqrt{57}+9}{2} x=\frac{9-\sqrt{57}}{2}$$