Solve 45-x=5/x

$$45 - x = \frac { 5 } { x }$$

Solve for x

$$x = \frac{\sqrt{2005} + 45}{2} \approx 44.888613177$$

$$x=\frac{45-\sqrt{2005}}{2}\approx 0.111386823$$

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.

$$x\times 45-xx=5$$

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

$$x\times 45-x^{2}=5$$

Subtract 5 from both sides.

$$x\times 45-x^{2}-5=0$$

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.

$$-x^{2}+45x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 45 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-45±\sqrt{45^{2}-4\left(-1\right)\left(-5\right)}}{2\left(-1\right)}$$

Square 45.

$$x=\frac{-45±\sqrt{2025-4\left(-1\right)\left(-5\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-45±\sqrt{2025+4\left(-5\right)}}{2\left(-1\right)}$$

Multiply 4 times -5.

$$x=\frac{-45±\sqrt{2025-20}}{2\left(-1\right)}$$

Add 2025 to -20.

$$x=\frac{-45±\sqrt{2005}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-45±\sqrt{2005}}{-2}$$

Now solve the equation x=\frac{-45±\sqrt{2005}}{-2} when ± is plus. Add -45 to \sqrt{2005}.

$$x=\frac{\sqrt{2005}-45}{-2}$$

Divide -45+\sqrt{2005} by -2.

$$x=\frac{45-\sqrt{2005}}{2}$$

Now solve the equation x=\frac{-45±\sqrt{2005}}{-2} when ± is minus. Subtract \sqrt{2005} from -45.

$$x=\frac{-\sqrt{2005}-45}{-2}$$

Divide -45-\sqrt{2005} by -2.

$$x=\frac{\sqrt{2005}+45}{2}$$

The equation is now solved.

$$x=\frac{45-\sqrt{2005}}{2} x=\frac{\sqrt{2005}+45}{2}$$

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.

$$x\times 45-xx=5$$

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

$$x\times 45-x^{2}=5$$

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.

$$-x^{2}+45x=5$$

Divide both sides by -1.

$$\frac{-x^{2}+45x}{-1}=\frac{5}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{45}{-1}x=\frac{5}{-1}$$

Divide 45 by -1.

$$x^{2}-45x=\frac{5}{-1}$$

Divide 5 by -1.

$$x^{2}-45x=-5$$

Divide -45, the coefficient of the x term, by 2 to get -\frac{45}{2}. Then add the square of -\frac{45}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-45x+\left(-\frac{45}{2}\right)^{2}=-5+\left(-\frac{45}{2}\right)^{2}$$

Square -\frac{45}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-45x+\frac{2025}{4}=-5+\frac{2025}{4}$$

Add -5 to \frac{2025}{4}.

$$x^{2}-45x+\frac{2025}{4}=\frac{2005}{4}$$

Factor x^{2}-45x+\frac{2025}{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}.

$$\left(x-\frac{45}{2}\right)^{2}=\frac{2005}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{45}{2}\right)^{2}}=\sqrt{\frac{2005}{4}}$$

Simplify.

$$x-\frac{45}{2}=\frac{\sqrt{2005}}{2} x-\frac{45}{2}=-\frac{\sqrt{2005}}{2}$$

Add \frac{45}{2} to both sides of the equation.

$$x=\frac{\sqrt{2005}+45}{2} x=\frac{45-\sqrt{2005}}{2}$$